Answer to your first question: yes - take a look at www.idos.cz for times. If you want to get the train all the way I am presuming you are going from Bratislava, in which case it will take c. 28 hours and will involve changing at Košice, Čierna nad Tisou, from where you get the world's slowest train (10 km in 1.5 hours) to Čop, from where you get the overnight train to Odessa.
Alternatively you could get the bus which takes the same amount of time - there are direct connections from Brno. A friend of mine does it quite regularly but he's welcome to it :)
Or fly Austrian Airlines/Ukraine International Airlines from Vienna, which is just down the road from Bratislava - c. 1 hour by bus.
Your second question - again I assume you are talking about Bratislava. You are only likely to be a victim of crime ONLY if you are getting up to no good yourself - stag parties frequently come under this category and therefore they are the usual target/perpertrators. Before they came along the city was a quiet small provincial capital minding its own business. You will hopefully not come across many stag parties a) because there are few places that will accept them nowadays and b) it has become too expensive for them. Crime is otherwise minimal and you can walk the city at any time of day or night with no problems whatsoever.
The only 'criminal' activity that you might come up against is that of the taxi drivers overcharging foreigners, but you won't need to get a taxi anyway :)
If you are talking about somewhere like Trebišov, then it is a different story ...