B1: Yes you can (are able to) reuse the wallet. But (as you already mentioned) the private key was on internet, so you can't be sure if noone else has it too. So if you want to be absolutely safe, don't reuse it.

B2: Yes and no. You will have to pay transfer(miners)-fee, so it will be less than 1 BTC.

B3: Website: I don't know, but you can take a look at https://en.bitcoin.it/wiki/How_to_import_private_keys . For example: you can install Mycelium on your computer to do the same thing on your own computer. This will be safer then using a website.

B4: No.

A1: I'm not aware of any website importing private keys, but you can do with a local wallet. Depending on the currency there are different possibilities.