Does my node rebroadcast its mempool transactions on startup? Say my node has transactions in its mempool and then gets shut down and restarted.

Does it save the mempool to disk and reload this mempool when it starts back up, or does its mempool get lost on shutdown?

If the mempool gets saved and loaded on restart, does my node rebroadcast these transactions out to the network once it starts back up?
When a node shuts down cleanly, it does flush the mempool to disk and reloads it upon startup. It will not rebroadcast the transactions.
A node will only relay transactions once, when it first learns about them. Your wallet will cause its own transactions to be rebroadcast. There is currently no mempool-based mechanism that rebroadcasts any foreign transactions after the initial relay.
Especially due to the privacy implications of this behavior (which makes rebroadcasts an obvious tell that the source is the sender), there is on-going work to change node behavior such that it will generally rebroadcast all transactions that should have been included in the previous block: [p2p] Introduce node rebroadcast module #21061