Is it possible to upgrade my Processor?

The only way you can upgrade a processor is if the CPU is fully compatible with your hardware and it the pin-out is exactly the same as the existing socket (and if your processor is socketed.) If you have a Quad-core model it cannot be turned into an 8-core.

You'd need to buy two identical Intel Xeon CPU's from the 55xx series. I don't know if the W5580 (3.2Ghz) will work (130 Watt), but the X5550 (2.66Ghz), X5560 (2.8Ghz) and X5570 (2.93Ghz) should work fine, as these are the same CPU's as Apple uses.
All three use 95 Watt, the 2.26Ghz (E5520 or L5520) uses either 80 (E) or 60 (L) Watt, depending on the version Apple uses (probably the E, as that one is ~$150 cheaper). So you might notice an increase in fan noise, as these faster CPU's consume more power and generate more heat (and therefore need more cooling).

In the beginning of 2010 Intel will release new Xeons, which, unfortunately, will use the LGA-1567 socket, instead of the LGA-1366 socket the current Xeons (and the Mac Pro) use. I don't know what will happen to the price of the older Xeons when these newer Xeons are released.

There have been some experimental bleeding edge willing to fork over the $2200 - 3500 cost, but not for faint of heart.

http://forums.macrumors.com/showthread.php?s=8153e865b4224f8fa24b427ba3d579c0&t= 781908

Wait for Gulftown.

YES! You absolutely can upgrade your Mac Pro! People will always caution you not to do it, but if you are good with your hands and can follow instructions, here's what you need to do:

http://www.anandtech.com/mac/showdoc.aspx?i=3597

For the fastest 8-core Mac on the planet, you will want to buy a pair of W5590 Xeons which run at 3.3 GHz. The cost for both of those Xeons will be... oh my... $1,700 x 2 = $3,400. That's a big chunk of change just for CPUs. If you re-sell your current CPUs on eBay, you can recover some cost. Be mindful of what is mentioned in the AnandTech article, as the stock CPUs in your 8-core Mac are lidless while the retail ones have lids (integrated heat spreaders). This will cause your heat sink to sit a few mm higher and you will need to make some additional modifications in order to get this setup to work.

Good luck!