No visa is required for Latvia, but you do need to get your visa ahead of time for Russia, and I think for Ukraine also (although I did once get one in the Kiev aiport...not sure if this is still possible). Check http://www.travisa.com/visa1b.htm for more info on all the requirements. Russian visas are a major pain. There is a lot of documentation to provide.

I don't think there is a way to book tickets online, although I could be wrong. It is easy to do it via a local travel agent, if you don't speak Russian and can't go to the station itself to do it. You can buy your Kiev-Moscow ticket in Kiev, and the two following trains in Moscow. You shouldn't have any problems with the trains selling out--there are many trains on these routes.

*Is it safe? --Yes. I have taken the train in Russia many, many times as a solo female. No problems. I travel second class, mostly (4 bunks per compartments).
*Do you bring an alarm clock or will they make sure you are off? I'd hate to sleep past my stop... --Bring an alarm clock just to make sure you get up in time, but you probably will be woken up by the conductor banging on your compartment door. They always want to take the sheets off your bed about 45 minutes before the train gets in.
*Can you get sleep on the train, is it comfortable? --I sleep like a log, but I am 5'2". The bunks are not large. I also bring earplugs. It can be hot in the summer--make sure you ask the conductor to open your window (it's locked at the beginning of the ride) when you set out.
*What are the accommodations like? Do they have a bed big enough for two people? --See above. Small, narrow bunks. No, there are not beds for two.
*Can you hop off and finish your ticket later if I wanted to stop in a town on the way? --Not in my experience. The ticket is for a specific train and a specific bunk, not the route in general.