There are four movies actually 1. Benizakura 2. Forever Yorozuya 3. Gintama The Final 4. Courtesan of the Nation (this one came out few months ago) out of these 1st and 4th are Anime Arc Adaptations! and the 2nd movie could be watched around episode 260 because thats when the movie came out, but it can be watched anytime during this season since it has got no spoilers related to future episodes
While the anime was airing, there were so many theories about what's under Takasugi's eye patch. I mean, eyes are like light bulbs in Naruto, and theories about Zoro's new scarred eye after the One Piece timeskip was making rounds too. So of course, the Gintama fandom was like "#1, Gorilla-sensei will copy these popular animes and hide something epic in Takasugi's eye. #2, It's gonna be a sharingan but only during a comedy ep. #3 Gorilla-sensei will hype us about something significant about this eye, only to troll us in the end" So... you may cast your vote or try and add a theory of your own.
There are three official movies. The 2nd one is an ending, since the anime was supposed to end and get its own ending while the manga continued. Of course, the anime continued, so it wasn't a real ending, though you can still make a lot of assumptions if you keep that in mind. The final movie is the real ending and basically an daptation of the manga ending.
8:54 On the stage Zura compared himself to Gian, one of the bully from the Doraemon series where they have different personality between the series and the movies version. Takeshi is actually Gian's real name and his mom often yelling out his name's in the series.