Can a threefold repetition be claimed in a blitz game? If so, how? Can one claim a draw by threefold repetition in a blitz game when no score of the game is being kept? If so, how? 

Once, in a blitz game, my opponent was simply trying to run me out of time. At a certain point in the game, I pointed out to him that he brought the same position on the board thrice and so I offered a draw, but he just ignored. There was nothing I could do to PROVE the threefold repetition occured and so had to carry on with the game and eventually lost on time.
It's hard, and I've never seen it done correctly, in the sense that it's the way I was taught it should go:

During slow games, once the players are not writing down their moves anymore because they are close to a game control, the arbiter should write down the moves as well to help with reconstructing the game after a flag falls (say to check if the 40th move is reached), or for claims like this. He can do so as well with blitz games, or appoint a temporary assistant to do so for him. It's just not usually done.

So you can stop the clocks and ask the arbiter to start recording moves, or appoint an assistant. In theory I think he should do that, and then when a threefold repetition occurs after that and you claim, the arbiter can check your claim using his notes.

But in practice, I think most arbiters will be too busy and too unfamiliar with these sort of rules to do that.

And of course, if you get the claim wrong, the opponent gets 3 minutes extra on the clock. Pretty hefty in blitz.