What is the incentive to broadcast the whole blockchain? What is the incentive to broadcast unprocessed transactions? Among the strengths of bitcoin is that it mostly allows all actors to act in their own best interest. There are some actions however of which it is not so clear to me what is the direct incentive other than some kind of idealism.

Obviously it is in a miner's best interest to immediately broadcast a newfound block, or blocks that validate blocks mined by him.

Is there a direct incentive to broadcasting the whole blockchain though (i.e. operating a full node)? Likewise, is there a direct incentive to broadcast all transactions to other nodes immediately? Naively I would think that it could be advantageous to a miner if other miners don't have access to some high-fee transactions.

Of course, for both it is necessary that enough nodes do it for bitcoin to function smoothly, but I am interested in the incentive for an individual miner.
Note that not all full nodes are miners, and that there are significantly more non-mining full nodes than there are miners (as in the vast majority of full nodes are non-miners).


  Is there a direct incentive to broadcasting the whole blockchain though (i.e. operating a full node)? 


The incentive for operating an archival node that can serve the blockchain to other peers is that doing so helps maintain the network and thus gives value to the Bitcoin that you are mining.


  Likewise, is there a direct incentive to broadcast all transactions to other nodes immediately? 


Yes. If run a full node which is not broadcasting transactions, then when you decide to make a transaction, it becomes immediately obvious to your peers as to where that transaction originated from. The incentive is to preserve your privacy.

Even with a delayed broadcast, you would still be risking privacy as your peers would like hear the transactions first from other peers, so as soon as they see you broadcast a transaction that they hadn't seen from elsewhere, it would be reasonable for them to assume the transaction came from you.