What kind of terms should an affiliate marketing program have for a subscription-based product?

1) It's absolutely normal for a business to pay commission on any one of the options you gave, here are some of the thought processes that go on in that decision:

• First, and in my opinion most importantly, what is the lifetime value of a customer to me? If I have a good (read statistically significant) estimate of how much money each new customer gives me on average then I can begin to at least put a lid on what I'm willing to pay an affiliate for a one-off commission. Meaning a commission once, after a lead conversion.
• Second, I can also think about the longer term commissions. If I know how long I can expect to keep a customer on board (retention rate or even better, churn rate) I can figure my profit from that customer and decide how much I prefer to keep for myself. This would allow me to provide a competitive commission while still ensuring I make what I want off each customer sent.
• Finally, it's important that if you do not know some of these values yet or don't have sufficient data that you choose a commission model that ensures you don't cash starve yourself.

2.) Again, this is a viable option assuming you can justify it with knowledge of your business. My suggestion would be to leave that option out unless it's a precedent set by competitors of yours already. I would much rather have paid my one-off commission or just continue at the affiliates converted rate. After all, if a customer "upgrades" a plan it is most likely YOUR work and success that led to that not the affiliates.

3.) Might need a little clarification here, but there are plenty of plug-n-play affiliate programs such as ShareASale and HasOffers that let you build one to tack onto an existing site.

I can't say enough that the point here is to HAVE DATA. Have an understanding of what you can afford or in lieu of good information just fail safe. Good luck with the app!