The preposition в (+accusative) is used with the verb играть ‘to play’ before the name of the game:
игра́ть в ша́хматы ‘play chess’
игра́ть в футбо́л ‘play football’
игра́ть в пря́тки ‘play hide-and-seek’
With musical instruments, на (+prep.) is used instead:
игра́ть на гита́ре ‘play guitar’
игра́ть на скри́пке ‘play violin’
You’re right.
Yes. Как пришёл Андрей here means something like ‘the event of Andrey’s coming’.
In English one can say e.g. ‘I saw you leaving the house’.
In Russian you cannot use a participle here (technically, you can, but it wouldn’t sound natural). Instead, you can say «Я ви́дел, как ты вы́шел из до́ма», literally, ‘I saw as you left the house’.
Correct. Well done!