Mail.app - Not retrieving automatically, flagged mails re-appear

I upgraded to the latest Mavericks with Mail.app fixes, and I continue to have problems. Specifically:

1. Mail is not picked up, or when it is, it is sporadically. I receive the mail on my iPhone / iPad, and I either won't receive it on Mail.app until I force it to, or it may take minutes to hours for it to happen automatically.
2. I cannot remove emails from any of the flagged folders. If I archive, delete, or unflag, it disappears from the folder but on clicking away and back, it reappears. The mails are no longer starred in Google Mail, but they persist on Mail.app.

Any ideas?