Let's break your issues into two separate cases: email and iOS update. They should be relatively orthogonal problems (not interacting).
Regarding email, it might help to know what provider you're using for email (gmail, Hotmail, yahoo, etc). A common problem of years back was that providers were switching from non-SSL connection methods to mandated SSL under IMAP; it's possible that you have an obscure provider that is just now getting around to removing non-SSL connection capability, or perhaps this is an iPad you haven't used in a while and now you find that your provider has changed their supported connection methods since you last used it. In either case, the solution is probably to delete the account on your iPad, and then add it back in. Apple tries to make the detection and configuration process for adding an email account as automated as possible, but it is possible that with an iOS version this old there may be a mismatch between what iOS thinks the provider wants and what the provider actually wants for configuration now that it is 2018. In that circumstance you'll have to look up things like server name and ports for your provider on the internet, and use the "Other" option when adding the account (other in this case meaning "manual configuration").
iOS update: your iPad 2 should support update to iOS 9.x according to everymac.com. The spinning beach ball you describe implies that you're trying to update via iTunes on your Mac. I think we'll need more detail about how far you get into the process before you hit the spinning beach ball, as well as what version of iTunes you're running. Also, when you connect your iPad to iTunes, does it successfully do a backup and sync automatically, as expected?
