I've heard of people doing it successfully, so it can be done, with some care and the right equipment. When the solder paste melts the chip aligns itself better than a fine-pitch QFP, so it can actually be easier than the latter. I'd be happier about doing it with an FPGA, as any missing connections could be corrected by rerouting the signals. The PCB has to be properly designed, of course.
I'd start with a QFN chip, they have some similarities to a BGA. If that works, I'd have some confidence that a BGA would be successful.
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.
Best Answer
The temperatures look about right; however, before you start try to achieve the 1C/sec dynamic over the area of the chip you are about to rework while maintaining accuracy, i.e., you need to be at 239C, then heat up 1C in one sec then stop at 240C. You will need a big heater with thermal mass much larger than your board (and preferably two - one of each side of the board), with good temperature control. Re-working large BGA packages is non-trivial, expect to fail the first time you attempt this so don't practice with expensive chips.
Pre-heating to 190 can be done slower, almost any rate will do as long as you don't stay anywhere close to this temperature for more than couple of minutes.
Also think about inspecting your work after reflow - you'll need access to an X-ray machine.