(1) It is straightforward for you to be named on the mortgage and the Land Registry Register. It can be done only through and with the agreement of your mortgage lender. They will call it "Transfer of Equity" and it will cost between �400 - �800.
(2) To achieve what you want you must own the property as tenants in common (TIC) with the shares in proportion - that is to say that if the property is worth �250000 and your partner's equity is �50000 then this would equal 5 shares. Your partner would therefore have 3 shares and you 2, which would be shown on the Land Registry form as 60% and 40% (a separate "witnessed agreement" by a solicitor is not required).
(3) You can both leave your TIC shares to whoever you like, so to achieve what you want for the son your partner would simply will his shares to the son.
(4) In the event of you splitting up your partner would simply purchase your TIC shares pro-rata to the value of the property at that time.
(5) The one thing that you cannot achieve is that either of you have an absolute right to remain in the property after the death of the other. Your partner's son on inheriting his father's TIC shares can apply to a court for an Order to sell which will not be refused, and similarly whoever you leave your TIC shares to can do the same.
(6) This could only be achieved by owning the property as joint tenants, in which case (2) - (4) above would not be possible.