Learn Russian: После войны он очень изменился.

Not quite! The current lesson focuses on the passive -ся added to imperfective verbs (like дом строится - the house is being built). However, изменился is a perfective verb. Here, the -ся suffix makes the transitive verb изменить (to change something) intransitive (to change oneself). So it means "he changed" as a personal state, rather than "he was changed" by an outside force.

Why is the word for war spelled войны here instead of война?

The preposition после (after) always requires the Genitive case. For feminine nouns ending in -а like война, the ending changes to -ы, giving us после войны.

The English translation says "a lot," so why does the Russian use очень (very)?

In English, we usually say someone changed "a lot" or "very much." In Russian, очень is the most natural and common way to express this high degree of action with verbs of change, emotion, or state. You can read очень изменился literally as "changed very much."

Could we use the imperfective form изменялся instead?

It wouldn't fit well here. The speaker wants to emphasize the completed result of the change (he changed and is now a different person). The perfective изменился captures this completed shift. The imperfective изменялся would focus on the gradual process of him changing over time, rather than the final result.