Is it possible to catch the train at Sestokai for Warsaw or do I have to return to Vilnius?
Definitely no need to go via Vilnius, because there is an everyday Sestokai-Warsaw train.
And anyway there is no more direct Vilnius-Sestokai train (connected to Sestokai-Warsaw one), not it starts only from Kaunas, but it still doesn't make sense to go via Kaunas either, since you'll then again come back to Sestokai by rail and have to change there for the polish train to Warsaw.
If it is possible, would you know the departure time, ticket cost and duration of this journey and whether I need to book tickets in advance?
Currently Polish train #10012 departs Sestokai 15:02 (Lithuanian time) and arrives to Warsaw-Wsch. 20:10, Warsaw-Centr. 20:20 or Warsaw-Zach. 20:29 (all Polish time), so total journey time is 6:18 till Warsaw main train station (which is 'Warszawa Centralna'). Direct ticket Sestokia-Warsaw costs app. 15 EUR (payed in LTL should be a bit over 50 LTL). You don't need to book anything in advance, seat reservation is not obligatory and anyway number of passengers traveling over the border is minimal, so no worries at all about free seats
p.s. just in case, there is also an everyday bus from Druskininkai to Warsaw (12:15-17:10) operated by SimpleExpress
. Standard full fare ticket here cost 12 EUR, but in case you purchase it in advance (like now for May departures), it's really easy to catch discounted offer and pay only 3 EUR for bus ticket Druskininkai-Warsaw