It probably varies from shop to shop, but I suspect most shops would allow you to use their rental gear, the same gear they give their Open Water students. An advantage of using the same gear, is it's less likely to confuse or distract the students.
Do you recommend acquiring a second set of pool specific equipment, or just a cheaper "throwaway" pool BCD?
If I couldn't use shop-gear for an open water (intro) course or refresher, I'd be concerned the shop was too cheap. More advanced courses may of course require gear that is not normally available for rentals. The shop obviously needs enough rental-gear for students, so what is one additional set?
That said, an advantage of having your own, is you don't have to adjust it, it's always available, you don't have to worry about fit, and you know how well it was (or wasn't) maintained.
I don't understand working for free, but if that is what you want to do, enjoy.
If I'm not making money teaching, then I would much rather be fun diving.
Teaching is work. Even if you enjoy it, it is still work and should be compensated.
100% this. The shop, dive-boat, or other business will be making money off of students one way or another. Even when a business does something "at cost" they usually do that to drive other business.
A couple months ago, I had to listen to a rant by a LDS owner about how he supposedly makes no money off of tank-service including fills, inspections, tumbling, etc. I get some of those things are probably not that profitable, however I know he was lying about VIPs, because I've been trained on that, and on most tanks that's practically printing money, especially when done in bulk and at the prices he was charging.
it may be worth keeping an eye out for a cheap back inflate BCD
You could always grab something off the used market.