Why would somebody steal her basket and hide it in her grandmother's mailbox? And how did she not notice it there?

Good question. Often wondered about it myself. KQ2 is probably the game in the series with the most illogical situations and solutions to puzzles. This is exhibit 'A'. How the Count's belongings came into Granny's possession is exhibit 'B' (if you don't consider the fan-remake's explanation, which I don't).

Best explanation I can fathom: Red is an absent-minded child, and she simply misplaced the basket and someone else (for the sake of argument, the kindly monk from the monastery or the antique seller) found it and delivered it to the mailbox.

In the fan remake, the count and grandmother were husband and wife. They were ordinary human beings (not vampires yet), the king and queen of Kolyma basically. But then the monks (who are werewolves in the remake) didn't like their leadership because he built a library, so the people of Kolyma became more and more educated and less interested in going to werewolf church. So the monks sent a vampire bat after the king and queen. It went for the queen, but then the king sacrificed himself for her and took the bite, and THEN he became Count Dracula (though they changed his name to an anagram in the remake). But, the woman still had a few of her husband's possessions. I assume the count had more than one cloak so he let his wife keep one, and as a vampire he probably didn't care about the ring so he let her keep it. That seems as plausible of an explanation as any.

Yup. Like I said though, I don't bother with the fan-remake's non-canon explanation. It makes more sense that Red simply misplaced the basket, and one simply has to wonder about the Count's lost belongings.