Water – The safety valve on boiler is really noisy

A fellow Briton (I'm guessing from your terminology) will have a better sense of what's normal and customary there than I do. Certainly sounds like the installers screwed up.

What sounds like your major heating issue is probably air trapped in the pipes/radiators - particularly the ones that do not heat well - there should be some sort of air vent on those (near the top - possibly on the pipe, possibly on the radiator itself) which you can open to permit the air to leave. Shut when water starts coming out. The expansion vessel is somewhat related, as it makes up the difference in volume from a cold to hot system so that the system does not blow off water (assuming there is a pressure relief somewhere.) If you open a bleeder and you don't reach a point where water comes out, the lack of an expansion vessel would be partly responsible - without one, system pressure will drop rapidly as you remove air, and may drop too low to push all the air out, unless you add more water to the system.

If you really don't have an expansion vessel and pressure relief, and the system is, in fact closed, that creates a significant hazard, as system pressure could rise to unsafe levels as the system heats. I would expect that the boiler would at least have a pressure or pressure/temperature relief. If there really isn't one, you have a potential steam explosion... or you don't have the system type you think you do.

I've not used a system like you have, but I'll give you some information based on what's common for heating systems in the UK.

As you've discovered, the immersion switch immediately switches the heater on. It's usually used as an override when you want water to start heating outside the timed period. (I'd hazard a guess that having the immersion on for 10 minutes won't be long enough to heat sufficient water for a shower).

You'd usually set the heating on timed, but it will take a while to heat up the full tank of water. For example, if you get up at 07:00 and have a shower, you might need to set the timer to come on at 06:00 to give you a tank of hot water ready for when you get up.

It's common to set the timer to come on twice a day, so you get hot water in the morning and again in the evening - though you'd adjust that dependent on your usage pattern. It might take a little experimentation to find out when you need to set the timer to come on to give you the hot water you need.

If the T&P valve is flowing quite an amount of water, it sounds like something is amiss with the system.