Yes. You can certainly tell people about the subscriptions, anywhere you want, but unless you're prepared to make changes to your store and product models, it's easier to keep it all separate.
Yes. It isn't impossible to add subscriptions to the store, but are you sure people will be looking at your online store for a subscription? If you advertise the store as a place to buy goods then they might not think it's the same place to buy access to the website. If it features prominently as a purchase place, maybe they would.
You could also not offer the subscription as a "product" but as something they opt into. You can keep the details of the subscription as a sort of hidden product that they cannot add to their cart - only your PHP code can do it, which it would when the user goes through your subscription page thing.
Treating the subscription as a product means you have to add a concept of conflicting products to your store. That might be useful for other things, I don't know, but if not then you're creating a lot more complexity for only one thing.