You can get pretty good (almost certainly good enough) privacy without running a node, just using feather wallet (or another) and connecting to a node over tor.
Wouldn't the same apply for Exodus or Electrum over tor?
If you do run your own node, you can just turn it on as necessary, but it might take significant time to sync depending on your CPU and Internet.
Do I have any say whether I want it to sync or not?
Running your own node is best for privacy, and if you do leave it on it helps strengthen the network.
If I decide to leave it on 24/7 is that with or without tor?
Re: exodus or electrum - yes but no. For electrum versus feather, xmr offers a lot of inherent privacy features compared to btc. For exodus theoretically I guess using xmr with tor proxy or whatever is same maybe, but exodus is a closed source wallet so you really can't know what's going on in there.
It syncs when you turn it on. It's not really a big deal, takes a few min to few hours. If you leave computer on with node running it stays synced.
You can direct network/block traffic and/or wallet/tx traffic over tor. I think commonly blocks are handled over clear net and txs broadcast via tor.