Why is Gintama's episode 305 full of contradictions?
Much like @Akaik mentioned in his answer, the main reason for the inconsistencies is entirely down to both Sunrise and Gori-Sensei not entirely knowing the details of these events as they go head first into making everything.

They pretty much admit to this using Gintoki to explain at the end of episode 306, where they include another Ginpachi-Sensei segment.  They even use the point you mention that previously it looked as if Gin joined Shuoyou's school after Katsura and Takasugi, when in the latest episodes it appears to be the reverse.  They make a joke about it not having a script at the time and that something entirely different is happening.  

This is probably as close to an offcial answer that we'll get.

There may be other mentions to why the other points you make out have changed in-between episodes but no mention of this (even as a joke) has appeared in the manga as far as I remember.