Learn Turkish: Cüzdanı açmışken çocuğa bütün bozuklukları verdim. - Since I opened my wallet, I gave the child all the loose change.

The suffix -ken means "while", but adding it to the perfective suffix -mış creates the meaning "while in the state of having done something". It is heavily used for opportunistic actions, meaning "Since I had already opened it, I took the opportunity to do this." If you used açarken, it would mean the literal ongoing action of "while I was in the process of opening."

Could I use açtığım için instead of açmışken here?

Yes, but you would lose the opportunistic nuance. Açtığım için simply means "because I opened" as a logical cause. Açmışken specifically implies "since the wallet was already open anyway" or "while I was at it." It shows you took advantage of a pre-existing state.

Why does cüzdan take the accusative suffix -ı here when the English translation says "my wallet"?

In Turkish, the specific wallet you own and are interacting with is a definite direct object of the verb açmak, so it needs the accusative -ı. The possessive suffix is often dropped in everyday Turkish when context makes it completely obvious whose item it is, but the word still behaves as a specific, definite object.

How does bozuklukları come to mean "loose change"?