OP: it sounds like you're still a bit confused about your rights/what to look for. I have lots of friends with English as their second language and when you get into legalities, different systems to those you grew up with...it can be really confusing.
So I know I'm repeating lots of good advise you've been given, but to summarise:
There are lots of different ways to jointly own a house with someone. You can be tenants in common or joint tenants. Joint tenants mean, if he dies, you automatically own the rest of the house. It isn't treated as part of his will, even if he changed his will tomorrow to one where he left everything to charity you'd still own his house. Tenants in common mean you each own a certain % and can choose to leave that % to whoever you want. Without a will I think that means it would automatically go in trust for your children. So if you do own like this, a will is a critical thing. Given his sizeable deposit and the fact you signed documents to ring fence it, I wonder is it more likely you own as tenants in common. I think that's an important thing to find out.
The second issue is around how you bought the property. To buy a house with a mortgage you need a bank, and a solicitor. The documents you sign with the bank are all just about how you finance the loan. The documents you sign with the solicitor are the ones that deal with ownership, and things like how you own. And you don't always have to both be at the solicitor at the same time to get all the documents signed.So the fact you remember signing lots of documents at the bank doesn't automatically mean you are co-owners.
From what you say, its most likely you do own a share of the property. But there are lots of different ways it could have been done. For example, lets say his deposit was 30%. of the property and the property cost 100 thousand - so he put in 30k. You could have signed documents to ring fence that 30k so whenever the house is sold he gets it back. You could equally have signed documents to say he gets the equivalent of 30%, so if you were to split in 20 years and the house was then worth 300k instead of getting his 30k he would actually get 90k.
These are details you really should know. Other people have suggested the land registry will tell you who owns the property. I think you could also see a solicitor and get them to find out, and explain the implications for you. It may cost a few hundred pounds, and I wouldn't tell your partner right now, but it would be good for you to have clarity.
In terms of all the other documents - yes, you can get him to add you to them. If he's hit by a bus tomorrow it would mean you aren't destitute. But if he has an affair in ten years, he can simply change it all. There's basically nothing he can't change.
So I think what people are advising is to plan as best you can for if the relationship would end. Now, you may choose to stay. You may think that getting to work part time is worth a few more years financial instability. But I would be looking at your own pension, and your own savings, and doing the best you can to maximise those. If you split in 5 years, and all you had was 50% of the equity in your house minus his deposit - would you be able to get a mortgage to buy on your own? Would you be able to support yourself?
Its not necessarily as black and white as 'marriage or you leave'. You need to look at what you can do to improve your financial situation, because aside from the issue of how the ownership of the house is set up, absolutely nothing he does short of marriage will hold any weight in the event of you breaking up. And realistically that is more likely than his death.