Do you have to connect your email to the same site you are hosting your site in? Also what are good payment processors to have besides Stripe and Paypal?

I will have my site hosted with AWS and would like to know if I have to use AWS to connect my business email address to. If not, what is a good business email account to use that does not cost much for a personal email address?

Also I would like to know besides Stripe and Paypal, what are other payment options I should have? I plan to sell within US (including territories and Puerto Rico), Canada and Mexico as of right now. Maybe in the future I will expand to other countries but right now my main focus is within the mentioned countries.