I can answer some of these...!
1. Can a car without MOT sit parked on the roadside next to my house with no issues?
No, I don't think it can. If it's a public road, it would be illegal for the car to be on it, even parked.
2. Can I transfer my current insurance and green flag to another car?
Yes, you can probably do that, although your insurance company will probably charge an 'admin fee' for that. (Mine charged £25 to change my address when we moved, for example.)
3. If the ECU ever does arrive, how can I drive my car to the workshop if no MOT or do I take my chances? Would the dealership store my car until the part arrives and how much do you think that would cost?
It would be illegal to drive the car to the garage with no MOT, unless you're literally driving it to a booked MOT appointment AND it hasn't previously failed an MOT on a dangerous fault. Whether they would store it until the parts arrive is up to them -- I'd be surprised if they did, to be honest, because most garages are tight for space.
4. Should I part exchange my car without a new ECU/due to fail its MOT to the dealership or local garage or private sale, or would you hang tight?
You can sell/px it but it would be worth a lot less compared to if it had an MOT.
5. Would you go onto using the bus in the meantime (might be doable but going to cause stress because single mum/swim with my two/have to be at work at 8.15/would need all shopping with slots) or use a hire car or buy a second, cheap run around to tide me over (if cheap, might have a myriad of problems)
That's a 'how long is a piece of string' question tbh.
6. How long can a car go unused without battery going flat?
It varies... couple of weeks, maybe?
What kind of car is it, and how old is it?