Double Emails

I have an iPhone 4S and a 13" MacBook Pro. I currently have 3 email accounts that are sent to both the Mail app on my phone and Apple Mail on my Mac. However, when I move a message from my inbox to an On My Mac mailbox on my Mac (duh), it is removed from the inbox on my computer but remains there on my phone. Ideally I would like it to disappear from both since it is now in a different mailbox. Any ideas on how to fix this?

Also, in a similar vein, the "Rule" I set up to filter junk removes all spam from my Mac Inbox (to the junk folder) but not from my iPhone (stays in the inbox). I assume there's a common problem here.

Thanks for the help in advance!