What should I do, as I stole a phone before reaching puberty and do not know its owner, who lives in another country? Should I donate its price on his behalf? If I do that, will it become mine?

Whoever steals before puberty is not sinful, according to the hadith: "The pen has been lifted from three: from the sleeper until he awakens, from the child until he reaches puberty, and from the insane person until he regains his sanity."

However, the lifting of sin does not negate the obligation of restitution. Therefore, you are obligated to return what you stole to its owner. If that is not possible, you should give its equivalent in charity on his behalf as a guarantee. If you later find the owner, you give him the choice between approving the charity or receiving the value of what was stolen from him.

As for owning the stolen item while paying its value, there is no harm in that.