No, there is no English requirement for the partner-sponsored temporary work visa.
Some people have got a visa while lodging with a family-member. A letter from your father isn't quite as solid evidence as if it was from some landlord who was a stranger, where your renting was just a commercial transaction, though. It is important that your father makes clear that you share private space as a couple - shared sleeping quarters, particularly - so INZ know that your partner is not just a separate lodger in his house.
Yes, that is fine. It is showing that an official organization is aware that each of you is living at that same address. Get as many more such letters as you can.
https://www.enz.org/forum/showthread...375#post574375
The plans are good. What will be better is to DO some things together, and show the tickets, photos that you are both there, etc.. Also, research some serious plans, not just leisure ones, for instance, checking out what housing you might want in the future, checking out possible savings plans to enable you to get to that point. (Even if it's not possible yet, you can ask institutions to write to you - both, or as individuals - about what is available.)
Marriage is not necessary to be considered partners.
No, that is not necessary.