Should Lithuania presently have the right to govern Vilnius?
Yes, very much so.Lithuanians have been living there for almost 70 years, ancient history doesn't matter.
Germany, through the auspices of the EU, invests a huge amount in Polish cities. They also have historical claims to places like Gdańsk. Should it be restored to its former name of Danzig?
If you go by "who was there first should rule there forever" than Germany can have no claims over Gdańsk.
Listen, there was no Polish or Lithuanian state prior to 1918 and everything was reset at that time (otherwise you should argue that 1914 borders should remain unchanged). The dominating doctrine at the time was self-determination* and if the inhabitants of Vilnius wanted to be a part of Poland, then they should be and Lithuanian government was the occupant no matter how you slice it. Here's what happened in 1918-1920 period:
November 1918 - Belarus
claims Vilnius and surroundings
December 1919 - Soviets conquer Vilnius
January 1919 - the local Poles try to fight Soviets back
April 1919 - Polish army's offensive is successful
June 1920 - Soviets launch an offensive, reconquer
15-20 August 1920 - Poland kicks Russians' butts
27th August 1920 - Soviets withdraw from Vilnius and hand it over to Lithuanians, effectively Lithuanians help Russians to shorten the front and give military support as Poland regains territories lost in June
September 1920 - Soviets lose yet another battle, Poland tries to negotiate with Lithuania to no success
October 1920 - in order to minimize to the bloodshed, Poland signs a treaty with Lithuania to keep their guard down, launches offensive the next day.
Really, Lithuania tried to f*ck Poland but lost at their own game. If Vilnius was Lithuanian, they shouldn't have entered the battlefield, simple as that. As they did enter the battlefield, they should have expected we would fight for it.
*didn't stop Britain from crushing Irish uprising in 1916.