1) driving is OK. like in the rest of the central and eastern europe))) it takes some 4-5 hours from BA to poprad, depending on the situation in zilina, strecno and ruzomberok. trains are fine, but now with the hystery aroubnd face masks in the trains i wouldn´t bother to use train...
2) download "mapy.cz" application and then offline maps for slovakia. it is great application which shows you all the roads, hiking paths etc. i wrote more about the the hiking in tatras in another thread in slovakia, just check it
3) yes it is. tatras are small enought that every trip can be done in one day. stay in poprad, nova lesna (there are a lot of private appartmans and family houses offering the housing and the quality is quite fine), gerlachog, svit etc. there is a train goind from poprad to strbske pleso via stary smokovec, as well as to tatranska lomnica, the connection is very well and the train is far the best for moving around as many hiking paths start at the train stations. also, you can use buses, they are quite freqent and they can take you to the areas not covered by trains (e.g. to "biela voda", where you can take the yellow path to zelene pleso, which is the hike you might like a lot).