"Purse" works as well, although a better translation is "portmonetka".

Well, one usually loses one's own wallet, not someone else's wallet.

Generally, English uses possessives a lot more than Polish does. That works especially for members of family, but also for objects. If it seems logical that the object belongs to the subject of the sentence, you don't specify it. If the woman lost someone else's wallet, then you specify it.

Also, "a" and "the" are accepted.

Well, it's how English works. In English you'd rather say "She lost her wallet", in Polish you just think "well, whose else's wallet could she lose?", so you don't need any word for "her", because it's obvious. Unless she really lost someone else's wallet, in which case you specify it.