Does anyone have info on the train number and what train it is, i.e. Ukrainain, Slovakian etc. as well as how to purchase tickets.
Actually it is not really a whole train, but just one single direct through UZ carriage, which departs Kyiv daily with train #99 (Kyiv-Uzghorod) at 15:22, then in Chop (from 6am till 9am next morning) it gets bogie change, border/customs control and reattachment to cross-border train Chop-Cierna nad Tisou. In Cierna once again it gets reattached (now to Cierna-Kosice local train) and continues to Kosice, where final reattachment happens (now to Kosice-Bratislava train). Arrival to Bratislava at 18:05, so it takes almost 28 hours for the entire journey
Carriages are Ukrainian, renovated not that along ago (think they all used to run in Kyiv-Berlin train before it was canceled), so basically it's the best thing Ukrainian Railways (UZ) have. Here the photo of Kyiv-Bratislava carriage taken just a few weeks ago
Ticket... well it's a bit tricky. Ukrainian Railways doesn't sell tickets for this type of international connections online, so the only way to buy it is at the station. However it's not that big of a problem, since these carriages to Slovakia pretty much always empty (partly due to the fact that it's expensive to travel the whole route with international ticket), so the situation when it's sold out is basically impossible.
Ticket fare for Kyiv to Bratislava depends on some additional factors (whether you travel one-way or return and whether it's just one passenger or more), but as a rough estimate for one-way one passenger ticket+sleeper reservation we speaking about sum around 85 euro (although when bought in Ukraine in UAH any time soon it can turn out to be even cheaper). But in case you want to save money you can travel using domestic trains this carriage gets attached to (--> getting off & boarding next one) for the total fare more than twice cheaper