If you are a miner and you "win" the block, yes, you can throw in any domains to the chain for free I suppose, and the network could accept or reject your block depending on whatever rules that network sets up to regulate that behavior.  That is like a Bitcoin miner can throw any transactions they care for as much as they care without having to pay a transaction fee, since they are paying for themselves.  Including a transaction fee in the process is just rearranging coins and doesn't change the net amount actually being spent or received.

I think that is of limited utility, unless you are saying that miners can eventually "earn" the "right" to a top-level domain after a certain number of hashes and inclusions into the domain registry.  That would be an artificial form of scarcity that would be interesting to implement, but it still fails to get fees paid by registrants to the registrar.  It would just make you "god-king" over a particular TLD... if you really "owned" that.

Such a non-coin "reward" also doesn't provide any incentive to maintain the database, while a coin-based reward would.