Well, first i would suggest you be careful in the openness of your relationship. its not what you think or believe, its the conservative opinions of the people you live around. The situation here reminds me of the former US military policy of "Don't Ask, Don't Tell!" I am not saying hide it, I am trying to say if you publicly come across as a "champion" of PRIDE, you may pay the price of public intolerance. This is your choice. I think most people don't care, but there are those that are threatened by anything different. i see same sex couples downtown, in cafes, at the beach. Never seen a problem except at protests.
you even as a foreigner can own property and a car in your name (together or you alone). Protect yourself and put your name on the title.
To stay legally you need to get some form of temporary residence. it starts by getting the first year, then it can be renewed for another 3 years, and then five years. Do you have some form of income? an easy way to get a temporary residence is to form your own business and register it. There are several immigration law firms in Kiev that can assist. I personally recommend a firm named B-Free. They have an internet site so you can start a conversation and get papers started long before you arrive
Another thing you should do is keep your current bank account and also get a Ukraine bank account. this way you can easily transfer money to yourself. Inside Ukraine they always want to know where the money came from. Large amounts of cash are assumed to be criminal. So I wire money from my left hand to my right hand and it is in the Ukraine system legally. You will find this necessary when paying for a car, a house, or even getting services. Almost all government functions require you to pay at the bank. It is archaic but it is the system (left over from years of socialist rule).
Good Luck
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