Personally, I'm using Touchdown for my work email and very happy with it.
To answer your first question, yes, if you set up your Outlook to use IMAP for your Gmail, when you delete a mail from your phone, it will delete from your inbox. If you log into Gmail via the web, click on settings at the top right, then Forwarding and POP/IMAP, you must first enable IMAP then you will see the link at the bottom for configuration instructions.