The Facebook messenger bot API seems to require you create a Facebook app for each bot, for each page. The Facebook app stores the webhook, and connects to the page.
Is there a way to have a single Facebook app for many bots connected to many pages?
If you create a bot to resell to clients as a service, you need to be able to do this. But there does not seem to be any way. You need to create a Facebook app for each client, and go through the Facebook app review process for each page which is very difficult.
How can this be avoided? Can a single Facebook app have multiple webhooks, or access to multiple pages messages?
You can create one chatbot and can use this bot in multiple Facebook pages. All you need is to store PAGE_ACCESS_TOKEN for each page and response messages to the user with right PAGE_ACCESS_TOKEN. You should get the necessary permissions to get page access tokens by Facebook Login. And then you need to subscribe the page to your chatbot(app): https://developers.facebook.com/docs/graph-api/reference/page/subscribed_apps
One important thing is Facebook generates a unique identifier for users for each page. Some user#A will have a different SENDER_ID/RECIPIENT_ID on different pages.