OK, lots of issues here, some legal, some not.
Firstly, if you intend to spend your life with this man, is it right for you to accept your mother's implicit judgement on him, and you, that it won't work out.
If you do not intend spending your life with him, why are you living with a man with a temper, who you are worried will walk out and whose fathering abilities you seem to doubt.
It seems to me you have a choice:
- Accept your mother's control and judgement, and her amazing gift, and accept that this may cause major problems between you and your partner as a risk you are prepared to put up with;
- Tell your mother that this is the man you are with and that she needs to respect your choice. If she can't then you cannot accept her gift (you wondered about the grown up stuff - this might be it/0; or
- Decide that there are serious issues with your relationship with your partner and sort that out first.
As far as whether or not you actually can prevent him from having any claim on the house, this is something that I think you will need to take legal advice about. It may only be possible if he is prepared to sign a contract, in which case he would need to be separately advised from you, but I am not sure about this. He may be willing to do this if it benefits his children and he gets a home practically mortgage free whilst you are together. That said, cohabitees do not have the rights that married people do.
Guardian - a number of issues with guardianship. Firstly, I would choose someone your age, rather than going up a generation. Secondly, I would choose someone who your partner likes and gets on with. Equally, if you want your children to continue to have contact with your side of the family, it should be someone who gets on with them too (and strong enough to mediate between them if your partner and mother do not get on).
Even if you give your chosen guardian parental responsibility, it is unlikely that your children would be taken from your partner if he survived you, unless there were good reason, like child protection concerns. But having parental responsibility would give the guardian legal standing in major decisions like schooling etc.
If the short temper / lack of stability actually amount to a child protection concern, then leave him. You will not be able to protect your children in the long term from that kind of domestic situation.
You ought to have someone named whatever, in case you both died together.
Once someone has agreed, you put them into your will. You can always change your will (but not rescind parental responsibility. Also, make sure you get their consent before you do this, and make sure that they understand what you are asking for.
Finally, make sure that the person you ask stays involved in your life and your children's life. If they barely know him or her, they will not be interested in forming a relationship with a near stranger when they are grieving, and it will take a huge amount of strength and confidence for the guardian to suddenly involve themselves with your partner and kids.
Hope this helps. Sounds like you have a lot of thinking to do.