if I put my EOS on Binance, they will register my tokens for me and I will be able to open up an EOS wallet afterwards?
Based on this announcement of binance, yes they will support for their costumers
https://support.binance.com/hc/en-us/articles/360001649171-Binance-Will-Support-EOS-MainNet-Token-Swap but doesn't mean it is free risk when snapshot of ethereum wallet happening because the error data can be happen, my own opinions will be better to register your ethereum wallet before EOS blockchain is available.
I think if you don't understand these technologies well, it's one of the safest ways to put them on the exchange. After that, they will automatically help you turn to eos wallet.
Of course, if you understand these technologies, your own conversion is the safest.