If you are using a POP account just make sure that you have your iPhone mail account setup to never delete from server. This way all of your emails will download to your mail program on your computer also making the need to e-mails to your computer it will download the same message. To change to that setting you need to click on your settings>mail>select your account>advance.
So this is the same email address? eg. ****@*******.com
I think you are probably using a POP3 server which deletes the emails of the server when ever they are downloaded to the device. You should change the setting to keep messages on the server an both devices and then you should be able to get all the messages on each device. The problem with that is that you have to delete the messages on both devices.
As to the forwarding problem, you are simply checking your mail on your iPhone before your computer, this takes the message off the server and onto your iPhone again. The above fix should help, or you should just not check your mail again until you have it on the other device.
NOTE: If it is set up with IMAP then you need to check the settings as all emails should be synced with all devices.