The webmail clients that we offer will only be able to check email accounts that are hosted with us directly on the server. When you use a Gmail account you can either POP in your email from various other email accounts such as the one that you host with us, and you can also use the Send Mail As
option to send out from your Paul@example.com
address instead of one ending in @gmail.com
You might find our guides on how to configure Gmail to access your email
and also how to send mail via Gmail
helpful in your situation.
Both of these are different than using Google Apps for your domain, which is a paid for service. You are simply choosing to use Gmail as your mail client, instead of RoundCube, or Outlook. If you send mail from within Gmail, those sent messages will only be available via the Gmail interface, and vice-versa for RoundCube. If you're setting up Outlook as an IMAP type of account, then it should be able to sync your sent messages across your devices.
Depending on the type of calendar invites you're receiving, I would assume that you would simply click on a link from the email to add it to your calendar application of choice.
Please let us know if you had any further questions.