Pro: Ways to offer members free periods?

Yes, you just create coupons and then send them the coupon code. A coupon will only work on the first payment made, so would be perfect for what you want.

I looked at the coupon code section but still am not exactly sure how to set it all up.

For example, if I want to give existing members two weeks free, how would I set that up? “SAVE-10|10%||ta-only” would only give trial accounts 10% off overall, right? Where would they even enter that coupon code if they’re already signed up? And where do I set up trial accounts, anyway?

To create a “trial account,” you’d create either a specific level or custom capability.

As I said, coupons give discounts on the first payment only. But you’re right that you couldn’t use them for a subscription that already exists. Rewarding someone for a reference is what affiliate links are for.