Usually another plugin has broken JavaScript on the page. Open your browser console and look for errors. A caching plugin that combines scripts is the usual culprit, so try excluding Quotly scripts from combining.
Send yourself a test WooCommerce order email. If that does not arrive either, it is your host, not Quotly. Install an SMTP plugin and send through a proper mail service.
The page needs the [quotly_basket] shortcode. If it was deleted, either add the shortcode back or pick a different page under Quotes, then Settings, then General.
Yes. Guests can send requests. An account only adds the Quotes tab in My Account.
No. They are separate. Adding to a quote never changes the cart, and nothing is written to your database until the request is submitted.
They stay. Deleting the plugin removes its settings but keeps your quotes, since they are business records. There is a filter for a full wipe if you really want one.