To ask what I should expect if I inititiate divorce?

[Zhan]

So say I tell him this weekend I want a divorce, what happens then?

House is joint owned. He put more in than me but that's not in writing anywhere. Who would stay in the house? Would it be sold? Could I stay in it and pay him off?

You need to see a lawyer - you can get a free half hour consultation with most solicitors. Then if you decide you want to go ahead they will write to your husband and say you have decided to initiate divorce proceedings. (You don't actually have to tell him if you don't want to)
Everything else gets decided afterwards, and that can mean lawyers every step of the way working out all the things you mention - who gets the house etc. Or you can decide everything amicable yourselves and then get it out in writing by the court so it is fixed.
Obviously if each point is contested this costs a lot more.
It's a long process with lots of paperwork, and all your finances will be gone through. Usually it's a case of you give a bit in one direction to get something in another.
It is also a fairly expensive business (mine cost £5,500 and it wasn't contested in court, but wasn't completely straightforward either)

You need grounds for divorce or he can refuse to divorce you (then you'd have to wait longer to get it) or you can agree that you have irreconcilable differences.

there's a lot of variables and it is complicated (even more so if you have children) so you need a legal consultation and they will advise you, and lay out your options.

Generally courts try to make the settlement equitable and fair to both parties (which doesn't necessarily mean everything is split exactly down the middle, it is also based on the needs of both parties)

Unless there is a court order, you both have equal rights to stay in the house. Neither of you HAVE to leave. That will be part of the negotiations.