Having viewed your latest pics, I have a number of suggestions/cautions that I'll list for your consideration.
Wear ridge at top of cylinder bores - if this ridge can be felt with your fingernail, it might catch the new stock pattern rings, so it'll need to be removed unless you opt for 'ridge-dodger' type piston rings. As I'm not a fan of ridge cutter tools, I'd opt for careful use of a curved Engineer's scraper to blend out any step, alternatively you might use your Dremel tool with suitable grinding bobs or rotary burrs. I think your cylinder bores need a bit more honing, I can still see shiny patches and some marks that I'd prefer to see addressed further.
I've been trying to figure what might have caused the weird pattern near the top of the cylinders, looks like sort sort of chemical erosion, it was very noticeable after you pressure washed the block (the pic of no.4 cyl. bore made me wonder if there might even be a crack in no.4 bore), but the erosion pattern is still present after honing. I'd be checking further.
New pistons often come without the rings fitted (so that you can check the ring gap in their respective cyl. bores) - don't be too worried about breaking any more rings, new rings are less fragile and it's easier to fit them than to remove them. Iirc, the Hayne's manual advises using thin feeler gauges or thin metal strip to aid in installing the rings in the absence of a proper ring expander tool. Suitable thin metal can be cut from cookie/candy/chocolate tins or canned veg - if you cut a section from a can a little smaller that the piston, you can slit the side and slip this over the piston to assist in sliding the rings into their grooves). Make sure the rings are fitted the correct way up according to the manufacturer's markings.
I still am of the view that your existing pistons might be fine to re-use and to fit them with 'ridge-dodger' type rings if available - this avoids any issues with the new rings catching on any remaining wear ridge at the top of the cyl. bore. I don't see the point in fitting new pistons in worn bores, better to re-bore the cylinders and then get the maximum benefit from the cost of the new pistons. This is an important decision and imho worth taking time to decide which is the better option.
(although you do seem to have made your decision and may even have ordered the pistons - I hope not, reference my much earlier comments about not rushing into buying any parts..
Cylinder block needs further preparation...
Have you removed the coolant drain plug on the block and ensured the hole is clear?
When I mentioned the coolant core/frost plugs, I was referencing them rusting from the inside out. If you haven't removed them I'd advise at least shining a light down the coolant passageways to get a look at their inner surface - don't be fooled by their outer appearance. (a borescope type camera might be ideal for this check.
Have you removed the crankshaft yet? Or did you intend not disturbing it?
Dirt is the big enemy of satisfactory engine work...
I would always remove the crank, even if I intended re-using the existing crank bearing shells. - The cylinder block needs to be properly cleaned - not just the coolant jacket/waterways/drillings but more especially the oil gallery and oilways/drillings.
E.g. there are iirc 6 oil feeds from the cyl. block to the camboxes, any dirt/grit that may have entered these will be pumped up to the camshaft bearings.
The crankshaft main bearings are lubed from the main oil gallery in the block, the oil that reaches the main bearings then has to flow through the main bearing journal oil drillings to reach the con-rod bearings. The oilways in the crank throws act like centrifuges and debris builds up inside the con rod journals - Fiat fitted core plugs (10mm diameter) next to each con rod journal, which are designed to be removed for crank cleaning during overhaul and especially after re-grinding and then new plugs installed.
There are also core plugs in the cyl. block main oil gallery that should ideally be removed for cleaning during engine overhaul.
If you don't want to mess with removing these oilways core plugs (and they can be tricky to remove without causing damage to the surrounding metal which can then make fitting new ones more difficult), then I'd strongly recommend a very, very thorough flush through of all oilways using a decent solvent. Old style 'pipe-cleaners' or small brushes can be used to clean out the crank drillings, try to pump solvent from the main bearing journal holes towards the con rod journal holes.
I am aware that the better solution to these pressed-in oilway plugs is to tap the holes and install threaded plugs, normally done on competition/ race engines but not really necessary for normal road use, I've never seen one pop-out in service unless removed and then badly refitted.
An engine can continue in use for many years without any problem while varnish, sludge and various debris can be accumulating in the oil gallery/oilways without apparent harm. But when an engine is left dormant for a long time, this debris can be loosened/dislodged when the engine is awoken and treated to an oil change, or better yet, an oil system flushing treatment and then can cause problems. Likewise, this debris will likely be loosened /disturbed if any work is carried out on the engine, so it's best to do a thorough clean while it's apart and not just assume it'll be fine...
Crankshaft bearing shells - I'd replace if any signs of wear e.g. surface coating worn through anywhere. It's advisable to pay particular to any pattern of wear on the bearing shells - can indicate other issues that'll need to be addressed. Don't worry too much about slight scratching.
OIl pump - can be checked out according to the Hayne's manual, pay particular attention to the condition of the splines on the pump drive shaft and check their fit in the oil pump drive gear that is driven off the auxiliary drive shaft. Check out the internal splines of this drive gear, they have been known to fail - I usually fitted a new oil pump drive gear unless the gear was perfect. Also check out the oil pump pressure relief valve, it's seating and it's spring. (I've sometimes fitted a spacer under the spring to reset the maximum release pressure, but don't overdo it).
Check the journals on the auxiliary drive shaft for wear, it's gear teeth for wear and it's bearings in the cylinder block - the bearing are replaceable.
Crankshaft - check the condition of the clutch shaft support bearing in the rear of the crankshaft - replace if worn or noisy (it may be possible to gently pop out it's seal and put a little hmp grease in to prolong it's life, then refit the seal. This bearing is often overlooked and if worn can result in a whirring noise when the clutch is disengaged, has confused people who thought the problem was the clutch release bearing or else inside the transmission. If you do need to replace this bearing (they rarely give trouble) make sure the one you obtain actually will fit the nose of the trans. front shaft before you install it...
Cyl. block and cyl. head missing studs - fit studs while engine is apart. If some studs came out when you undid their nuts, remove the nuts and fit the studs - don't do as others often do and refit the nut and stud as it was removed, this will likely result in the stud not being at full depth or tight in the casting and will likely loosen in service, on alloy castings, it can result in a pulled thread when tightened fully.
Cyl. head - I don't like the look of the Helicoil insert you've installed... I suspect you inserted it from the combustion chamber side? The exposed thread represents a sharp edge that can possibly glow red-hot and potentially cause pre-ignition in the cylinder. I'd either re-do it or carefully prise the lowest thread out of the threaded hole, notch it with your dremel tool (it's probably to hard a material to snip with pliers) to remove one coil and then pop the end back in place. Iirc, the end thread should be approx. 1/2 to 1 thread from the end of the hole so that the insert end digs into the surrounding metal to prevent it possibly being wound into the cylinder when refitting spark plugs in future.
Cyl. block - clean out any waterway drillings with a drill bit to remove corrosion. Clean out any threaded bolt holes (10 of them) to ensure no corrosion present and correct torque can be applied when reassembling the engine. Also clean the cyl. head bolt threads for the same reason. ensure no liquid e.g. oil in present in the holes before refitting the cyl. head to prevent a 'hydraulic lock' situation. Check the bolts for 'necking' (stretching) both where the thread starts and along the thread.
Cyl. head - ensure the holes that the cyl head bolts pass through are clean, you don't want any rusty particles being dislodged and getting on the gasket sealing surface (I've seen this happen on 'quick' head jobs, but you seem to have this already covered in your cleaning).
You mention some of the new valves were loose in some of their guides? I wasn't aware you were replacing the valves, I thought you were just going to clean them and then lap them in the check them...
One of your pics of the cyl. head shows a cam follower and shim in the background - that shim looks 'dodgy', I'd replace any dubious shims, also check any used shims being re-used are still perfectly flat, - hollow shims can quickly destroy a cam lobe.
Your pic of the cylinder head shows all valves in place, I'm guessing these are the new valves? And you haven't yet fitted new valve stem oil seals? I would only fit the valve stem oil seals after all valve grinding/lapping work has been completed and everything has been cleaned - this avoids the risk of any grinding paste grit getting trapped in the guide or under the new oil seal. Iirc the lower spring seats (a flat hardened washer topped by a stepped washer/ring, have to be fitted before the valve stem oil seals are installed, they can't be fitted afterwards, won't fit over the valve stem oil seals. Be gentle removing the old seals (I just twist them with a pliers) and fitting the new ones (should just push into place using a suitably sized tube or socket and maybe a gentle tap).
Check the condition of the valve collars and their retainers (collets). Try 2 retainers on a valve to check they fit correctly Check your retainers are the correct type for the valves you'll be using (some valves have 1 round groove, others have 2 rounded grooves, others have 1 deep square groove). With 2 retainers test-fitted on a valve, there should be small gap - when installing a valve, I try to get the gap equal on each side of the valve stem.
Check that none of the retainers have started to 'flatten out' or are showing signs of descending into the valve collar, i.e. test fit a collar and retainers on each valve (with the valve out of the cyl. head) and check that the valve stem is protruding the same amount on each valve.
Having said all this, Fiat twin-cam engines are pretty robust in the valve train area and give little trouble (unless of course, the ignoranti have previously been in there...).
The design of the Fiat t/cam valve operation is one of the best around, the use of cam buckets removes any side thrust on the valve stem leading to much reduced valve stem/guide wear in service.
I wouldn't worry about a little play, the oil seals should prevent excessive oil being drawn down to valve stem and then being burned - the pics of your engine as you disassembled it show no signs of excess valve stem/guide wear or consequent oil consumption.