I am not a MH owner, but as you have had no response, firstly there are many MH (Motorhome owners clubs on line who will advise.
Secondly and only based on a couple of caravans in the past and common sense , so make your own mind up about my advice.
In your listed order. Electrics you have the mains 240 volt when "hooked up" at site, you can test this at home with adaptor to see if everything works, often there used to be something I think called a ZIG unit which directed the 240 volt power to work your fridge etc. when on site and also to charge any leisure batteries. If using to run the fridge you may have to turn a switch to the correct position, some have mains/12 volt /gas.
With engine running you need to know if it is charging the vehicle battery and any leisure battery, the ZIG unit should control most of that if working correctly. Obviously check all vehicle lights etc. work and engine oil and other levels are OK.
Regarding gas this should be checked regularly by approved professionals for safety, however common sense checks would be to make sure you bottles are full before trip then when connected that there are no leaks anywhere in system, you can use your nose to smell, hearing if a bad leak hissing and what is useful is a container of water and washing up liquid to brush around all connections, this will show up bubbles if leaking. Professionals will have tools to measure the pressure if it drops and other devices. Check any gas water heaters and cookers, fridges etc. turn off fully at their controls. I would turn off the regulators on any gas bottles when not in use at the time.
Regarding operating gas fridges , the ones I have used had a pilot light which you had to check through an inspection hole once the igniter had been worked correctly with the gas tap in correct position, if pilot goes out and fridge is faulty it can be dangerous!!!
A carbon monoxide detector is a good safety precaution as heating appliances in a closed space can build up poisonous gases which can silently kill!!! Another reason for good ventilation at all times.
Regarding water I would thoroughly flush all systems, then for drinking water etc. maybe use some cleaning agent, possibly Miltons baby bottle cleaner so safe for humans but should kill bacteria, especially in old water containers that are left standing for long lengths of time, then fill fully before use as they soon run out, so if on site fill regularly.
Another thing to check is that all the water taps turn off properly with no leaks as if electric switches turn on pumps in the middle of the night it will wake everyone, including neighbours
Regarding grey water tank, obviously make sure what ever toilet system you have fitted works correctly and that again no leaks
also have it emptied before your trip, it is surprising how quickly they fill, so if possible use public facilities when available.
I recall several of us went on a canal barge holiday and half way through had to have it pumped out at our cost, although we suspected that the tank hadn't been fully drained by the operator!
Finally if engine battery is used to work leisure side you may find after watching TV all night you have to pay a garage to start your vehicle
If I have missed anything I trust others on Forum can advise.
