Does a synapse unit instigate battles they are at, if not at 1st planet or a planet with a warlord?

Yes they are considered a second Warlord with all that title represents.

Actually, no. The Savage Warrior Prime SPECIFICALLY initiates a battle, as per its card ability text. This is NOT an ability inherent to all Synapse creatures.

They are definitely not considered Warlords, they are a distinct type of units (synapse units, obviously). They do not win command struggles on their own, they do not initiate battles (or take initiative) on their own, they can be targeted by effects specifying non-Warlord units (like Zarathur's Flamers and Eldorath Starbane) and they cannot use a combat turn to go back to HQ.

Thanks, that's how we played it.