What would happen if someone goes and constantly generates new addresses as some kind of attack on the network? Would they eventually usurp most possible network addresses or get some other person's address and potentially usurp their payments?
No.
There are 1,461,501,637,330,902,918,203,684,832,716,283,019,655,932,542,976 possible bitcoin addresses.
If your calculator can handle numbers that big, you can play around with how long it would take to try generate one quadrillionth of them if you could generate a trillion per second.
(I get an answer of a bit over 46 trillion years)