Do miners do this to try and get a head start on finding the next block and getting a reward? Do these empty blocks live on the shared blockchain if they don't become stale?
Yes and Yes.
Since mining in Proof of Work is based on luck, they could get lucky in that few seconds and find a new block to get the reward before others do which is why they do it.
You can find such blocks on the blockchain, here is a block explorer link you can use to see said blocks that are empty (only have the mandatory coinbase transaction):
https://blockchair.com/bitcoin/blocks?q=transaction_count(1)#