When you use 把 in a sentence, you cannot have a simple verb like 写; there has to be something attached to it to make it complete (here 下来). (Chinese grammar books have pages and pages on this specific topic.) As for 您, it is the polite form of 你, mainly used with older people, or people you want to show respect to. Also used a lot in grammar book examples. Hope this helps. 加油！

您 把 您的 名字 写下来 = write your name

您 is a respectful call,the meaning is the same as 你，but 你 is a normal call