Would missing check valve make it seem like heat is always on

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.

Assuming the thermostats are working, you may be getting residual heat once the thermostat turns off.

Do the pumps have speed controls? Many pumps have LOW, MEDIUM, and HIGH settings. A lower setting may avoid spikes in temperature.

Also, many pumps have mixing valves near them that control the temperature of the water being sent to the radiators in that zone. If you have selective zones that are too hot, you may be able to temper (in this case, lower) the water going to that zone.