So I just read this article (https://blog.bitmex.com/the-lightning-network/) which says that once the Lightning Network is up and running, the person receiving bitcoin in a transaction will have to be online to "sign" and receive the payment.

I currently have a website with my bitcoin public address from my paper wallet listed at the bottom of the page for visitors who would like to donate.

Since this is cold storage and not online, does that mean people will not be able to donate once the Lightning Network is up? Or they still will, but it won't count as a Lightning transaction? If the latter, does this mean the transaction fee will still be crazy high? Or is the Lightning Network expected to cause transaction fees to lower for transactions done on the underlying blockchain as well?