Hi,
I toured Ukraine and the Crimea in 2010...I had a great time, and it's highly recommended. It can seem a bit sparse on attractions, and any medeaval heritage, as most of it seemed to have taken a battering in the war. But despite that I still enjoyed it, and it was certainly an adventure.
As regards the police....
I travelled around as much of Ukraine as I could (75%) so I covered a lot of miles, then went down to the Crimea and drove all around there. So all in all a lot of miles.
In that time the police were ever present, and I was pulled over many times.....
.....it did become a bit of a chore in the end, and sometimes the initial amount they want is just crazy!
But like has been suggested I took a second wallet with some Ukraine money and some dollars in it. But don't keep all you take in the wallet....drip feed what you have into so that when you are stopped there is very little to take....as a rule of thumb they are happy to take what you have, they'll settle for $20 if that's what you have, if you have $100/$200/etc that is what your fine is!!
They don't really want the Ukraine cash, and love dollars....on one occassion they turned their noses up at £40 worth of Ukraine money to take $20!!
Like I said it can become tedious, but it doesn't spoil your trip. If you handle it right it quickly passes, and they were all civil enough, it's all part of the game, and once you're used to it you can just play along.
I guess in the miles I covered it didn't happen very often per mile....and there was days when I wasn't stopped at all....but on the days when I was stopped 3 times, and given more to the police in a day than for my lodgings, then like I said it can drain you.
Also, now I think of it I wasn't stopped once in the Crimea!.....but I do remember that the driving in the Crimea was a lot lot worse!...horrendously bad!!....and the really bad accidents were a daily sight!!
I think it was a mixture of the Russian and Moldovian driving??
As for cost per day?...screw your loaf and each stop might cost you $20? You might not get stopped for days, you might get stopped three times in a day?....if you drive like an idiot, or very fast, then I think your risk would be a lot higher, and maybe your fine too?
Overall, having said all that I wouldn't worry too much...if you've riden in Eastern Europe I'm sure you know the score...it just seemed slightly worse there, but not excessive.
Go, and enjoy!