When I'm cleaning my bike, the quickest way to degrease my chain is to remove it and degrease it in a small tub. Will constant removal / re-adding to the bike cause damage? I don't have a quick release chain.
Depends on the chain. If it's a chain with cylindrical pins (most singlespeed and some narrow chains) then the damage will be negligible. Even if you always use the same pin you should be fine the first 10 or so times.
If it's a chain with profiled pins you'll need to use a new replacement pin every time and the replacements can't readily be removed (doing so damages the side plates). So you'll be limited to about 100 chain-cleanings before your chain is entirely made of replacement pins. And you will have spent more than the cost of a new chain buying pins.
There's two things to look for: damage to the side plates, especially dishing around the pin; and a pin that is too easy to remove (it's loose). Either means you're likely to have the chain fail in use. Replace that link.
The easy way is to buy a quick release link and use that. They're generally only good for 20-50 open/close cycles, but the failure mode is that they become steadily easier to open, making it easy to tell when you need to replace them. And they're fairly cheap - typically $5 or so each, compared to about $1 for a single-use replacement pin.
Pushing pins out and then jamming them back in are bound to cause a certain amount of strain on the links. Buy a connector link and replace one of your links with it, creating a "quick release" chain. Make sure that you get a link that is the same width as your chain.
Even better, buy a few of them. I actually keep one with me, that way if I break a chain it makes repairs so much easier.