Hi 민윤기! Let me try to provide you with some answers.
1) 请问您要办理什么业务？ means "Pardon me, what business do you hope to complete?" Word for word, it says "Please ask you want complete what business?"
2) 我想把欧元兑换成人民币。 You cannot omit the 成 because of the 把. For more information about 把, see here: https://en.wikipedia.org/wiki/B%C7%8E_construction
3) 他申请了三所大学。 (Literally: He applied to three universities.)
4) 签字 and 签名 are pretty much interchangeable. Both can be used as a noun or a verb. 签章 however is less frequently used and can only be a noun.
5) There isn't a significant difference between 为 and 给 when used that way. 给 does provide a more direct, intimate feel than 为 though.
6) You don't need 并且 there, but you have to rewrite that sentence. You may say: 虽然我来自波兰，但是我的韩语和日语都说得流利。
7) Correct, except out of habit, native speakers wouldn't put a 的 after 金 in this case.
Please let me know if I didn't answer any of your questions.