Honestly, I wouldn't try to solder my own BGA's. I know this doesn't directly address your issues, but hear me out.
It takes a lot of work and effort to solder a BGA. There's a lot of trial and error. A lot of messed up test boards. But then it's soldered. Now what?
Now you have to prove that it's soldered correctly. For that you need one or more of the following: JTAG test (US$10k, never has 100% coverage), optical inspection (US$20k for the equipment), or X-Rays (US$500k). The cost of doing these tests is too much for the normal hobbyist, and is even beyond many small companies.
Skipping those tests, you proceed with debugging your PCB. And let's say that the BGA is a complex CPU. Inevitably you'll find a bug. The CPU will randomly crash. Is it your software, your electrical design, or the soldering on the BGA that's causing the problem? Debugging this, in light of some possibly problematic soldering, is going to be terrible. It will add a lot of time to your debug process, possibly months, and you'll loose a lot of hair on your head. And then you can repeat this for the next major bug.
Without confidence that your soldering is perfect, you will always have this dark cloud over your head. Every little bug that shows up "could be a BGA soldering problem". This is made worse if you have multiple engineers working on the same PCB since the software guy will be questioning the hardware guy, etc.
Then, even if the BGA soldering is perfect, did the chip get too hot? Did you destroy the chip by getting it too hot? Even on modern assembly lines this is an issue. But with the proper equipment you can adjust and measure the temperature profiles to at least get you in the right ballpark. On one board I did recently, the BGA's were being damaged. The solder balls looked great, but under a very nice X-Ray machine we could see that the gold bond wires melted from the heat.
I've been there. Not at the hobbiest level, but professionally as we were bringing up new boards while the assembly shop was learning to do BGA's. We had no JTag. No optical inspection. And the X-Rays were terrible. Our PCB had 11 BGA's on them. That was 2 years of hell I don't wish to repeat.
So, here's my recommendation:
Get someone who has the proper equipment, training, and experience to solder your BGA's. There are a lot of contract manufacturers that'll do a single BGA. It takes money, but that's way less than the time you'll spend trying to debug your own soldering.
If you must do it yourself, then you should get the proper equipment, training, and figure out how to get the experience required. For this to pay off in the end, you need to have a large enough company and need to justify the huge amount of time and money that you'll put into this.
But I would never try to just kludge something together. That's a recipe for, um, bad stuff.
Could this possibly damage the board?
Of course.
What about all the parts?
Yes. It could certainly damage them too, and most probably would.
Are there any other pitfalls that I might encounter or would this method work well?
The problem with this method is that it is inherently poorly controlled. It is potentially able to work and an exceptionally skilled and experienced and well trained operator may be able to achieve somewhat acceptable results some of the time. But most of us would just end up with a work of art or a pile of smoldering slag.
Probablility ~= 1.: Reflow soldering is an exercise in controlled death. Components and board are heated up hot enough and long enough that they are well on the way to destruction. Manufacturers design parts to meet the stresses of this process with an acceptable margin of safety. If you read up on the reflow process in detail, as you MUST have already done to make this question more than idle time wasting, you will have found that temperature profiles - rates of temperature change, holding times and cooling times and temperatures are all tightly specified. If you can manage the sort of control that this implies over the surface of a PCB containing 250 or so components including fine pitch TQFPs then you are wasting your time in your present role and probably want to enroll as a micro-surgeon or Formula One driver or similar :-). ie it's far too demanding a task for this to have any certainty of working.
Probability ~= 0: Not everyone is Wouter - he is an extremely experienced and capable engineer. All that said, it is "just possible" [tm] that a consistent approach, well aligned jig, temperature controlled air source etc may be able to do the job quite well. Finding out could be expensive. Or not. Given the very great success achieved by the toaster-oven-PCB-assembly community and the large amount of on-web information available on this method and the relatively low cost of doing it, I'd expect your TQFP's to thank you profusely for taking that route.
Related:
Spark Fun show you how to do Toaster Oven PCBing - lots of details - MANY photos
Some amateur results
Lots and lots and lots of PCB-toaster-oven ideas
Open Hardware PCB toaster over project
And more ...
Even a small BGA - an instructable
Best Answer
First of all, if you are using lead-free solder, it sucks, if possible use leaded solder.
There are two categories of soldering. With interaction and without. Interaction refers to the necessity of having to touch the chip in any way during soldering.
The only way to solder a chip like this without interaction is to use a stencil and solder paste. The stencil dispenses the correct amount of paste. Most importantly the paste is even on all pads. It also almost eliminates solder bridging because the stencil separates the paste on the pads. It looks like you have a single side load. In a case like that, I like to heat the board from underneath, it keeps little chips from blowing away. If you can't do that, then heating from the top is ok but requires a bit more caution. The even solder paste pays off because the chip sits down evenly.
The way I suspect you are soldering the chip is that you apply a strip of solder paste on each side of the chip and then letting the surface tension pull the solder into place. This type of soldering will require intervention. Dispensing just the correct amount of paste is nearly impossible. The tendency is that too much is put on. This causes shorts that need cleaned up after wards. Also, it's suggested that once the solder paste melts, make sure the chip is positioned and then hold the chip down with tweezers and remove the heat. Hold the chip with the tweezers until the solder cools. This will force out any extra solder from under the chip keeping the chip from being crooked. Finally, use a soldering iron with a small tip or knife tip to clean up any solder shorts. Be sure to add flux before removing the shorts.
Another method is similar to above but tin the pads first instead of using paste. Make sure to add a generous amount of flux before putting the chip down.