As long as there is no sharding solution available, a fullnode must handle a tps equals to the tps of the whole network.
So if all participants send 1000 transactions per second, a full node should be able to handle 1000 transaction per second. Otherwise, the fullnode will be desynchronized and will not work as expected (mainly it will select tips that aren't tips anymore and so transactions attached by this desynchronized node have less chance to be confirmed.).
It is true that all fullnode don't need to have the exact same view of the tangle, but they must have almost the same view (dead branch don't need to be known by every node, but the main branch must eventually be on all fullnodes. Intuitively, it is the same in bitcoin : the longest chain must be known by all fullnodes, orphan blocks can be ignored).
Today, the tps on the tangle is between 3 to 8 tps, so fullnodes should be able to handle 3 to 8 tps.
Regarding the gossip : when a fullnode receive a transaction that isn't yet in it's database : it will broadcast the transaction to it's neighbors. If the transaction is already in it's database it will simply ignore it.
Regarding prioritization : the current IRI implementation order all it's internal queues according the pow invested during attachement. The min weight magnitude is currently 14 on the main net: all transactions with a hash ending with more than 14 zero trites will be handled in priority. (and of course all tx with less that 14 zero trites will be considered as invalid)