Yes super easy. You can do this using Gmail's filter service. However, many services will also let you log in to gmail via POP/IMAP (OAuth for auth) and fetch the email from your gmail for you that way.
> 2. When I send email from new account (custom domain), how are you completely sure it is not going to be on spam? I send unsolicited emails to people in big corps (journalist asking for interviews/comments, not spammy :-)
I guess it's hard to know for sure, but I've not noticed any problems with this when the underlying provider is a well-known sender such as Gmail or Fastmail.