How does the lightning network handle changing IPs? What happens if suddenly my IP/Onion address changes? Will that be a problem for the lightning network? Or could my node just simply update the network graph as long as its public key stays the same?
If your IP address changes, you reconnect to the network and broadcast a new node_announcement message containing the set of your node's reachable addresses and an updated timestamp. The node's public key identifies the node  and all of its channels and may not change. All other information in the node_announcement message can be changed.