The Core I7 will not work. The Dimension 8400 series do not have LGA 1366 (the socket for the I7) nor do they support DDR3.

It is possible (let me stress possible) that your mobo will support the Q9550, further information would be necessary and new bios might be required.

As you already noticed, your vid card is outdated and would probably be your easiest (most reliable) upgrade. Your board should have a PCIex16 1.0 slot on it (judging by your chipset). So there are a lot of options available. Your 7900 is the strongest card in your deck.

edit: beware, it has a 350w power supply which may not be sufficient if you make too many upgrades.