Would there generally still be any snow left in April to ski on, either in Zakopane or over the border in Slovakia?
I would think it is highly unlikely in Zakopane that there will be any snow left in April to ski on, and the only time I know of was back in 2001 because I was there then, but that was an especially long winter.
You will just be able to have some skiing in April on the High Tatras on the Slovakian side namely Strbske Pleso and if you go to this page it has a history of webcam shots where you can see the conditions by day for each month for the last five years.
If you want to know where to stay PM me, as I have a very cheap address in a restaurant very close to the ski run, but I am not going to post it here as it is too good a place to lose and would ruin my own chances of booking there.