Could the bitcoin network be destroyed by someone generating endless bitcoin addresses nonstop?  I mean it sounds like it would do something really bad to it.

I'm pretty sure at some point a website will take payments in bitcoins and will decide to generate a new bitcoin address for each payment and end up endlessly generating new addresses like mad and do something bad to the network.  Then more will start in.

Yes.

Also, generating a new address for each payment is the best way to do it. Maximum anonymity because of least re-use of addresses. The keyspace is large enough that we don't need to worry about exhausting it.

When you generate a new bitcoin address, it only takes disk space on your own computer (like 500 bytes).  It's like generating a new PGP private key, but less CPU intensive because it's ECC.  The address space is effectively unlimited.  It doesn't hurt anyone, so generate all you want.