GDANSK TO LITHUANIA

First let me please make it clear that I am well aware of the lack of railways in the Baltic states and the need to make use of buses to pass through those countries.

I will be leaving Brno in Czech Republic on 15th or possibly 16th July with the ultimate intention of making my way eventually to Tallinn in Estonia, but because I am very keen to spend a few days in Gdansk (Poland) en route I have been trying to work out how to travel by train from there (Gdansk) to Lithuania on or around 18th or 19th July with either Kaunas or Vilnius as my destination, from either of which cities I would then take an onward bus to Riga and eventually Tallinn.

I have a continuous one month 1st Class Interrail Pass, and the planner in the app informs me that it cannot find trains from Gdansk to either Kaunas or Vilnius. Nor can it find trains from Warsaw (Warszawa Centralna) to Vilnius. I presume the reason for this is that the Baltic states have not yet bothered to submit their railway timetables to Interrail, as has been mentioned on here many times before.

Nevertheless I have to assume that it is indeed possible to travel by train from Poland to Lithuania because The Man in Seat 61 website indicates that a daily train departs Warsaw at 07.35 and arrives at Kaunas at 16.25, (and at Vilnius just over an hour later). 

However, because I will be in Gdansk I will not have time to take this train from Warsaw without staying overnight in Warsaw the previous day - something I would hope to avoid, and so I am wondering whether or not this quote from The Man In Seat 61 website still holds true: ”but in June 2016 a cross-border Bialystok-Kaunas train started running over the new standard gauge Rail Baltica line.”

My first question therefore is as follows: As far as anyone knows, is there still a train running between Bialystock and Kaunas, as referred to by The Man In Seat 61 (see above), and if so, is it possible to travel by train to Bialystock from Gdansk?

Second question: Is it possible to travel by train from Gdansk to Lithuania by any other rail route, or is my only option to take the daily 07.35 from Warsaw to Vilnius (which passes through Kaunas, from where I could catch a bus)?

Thirdly, I believe that I need to make reservations when travelling by train on Polish territory. What is the best way to do this and is it necessary to do so for the Polish leg of the journey from Brno (Czech Republic) to Gdansk?

Thanks in advance for any advice.