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?