Can a node broadcast a valid signed raw transaction while still syncing?
You can only broadcast transactions if your node has synced to the point that it knows the outputs that were spent. This is because your node will always validate a transaction before it broadcasts it. If it has not synced to the point that it knows the outputs that your transaction spent, then it won't have the information necessary in order to validate the transaction.

So if your transaction spends outputs that were recently confirmed, then you will have to wait until it is basically fully synced before you can broadcast. But if your transaction spends very old outputs that are deep in the blockchain, then you can broadcast before it has synced.