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.
C: definitely the assembly shop, if you have the wallet for it. That's for you to decide. Ask some quotes, and decide if doing it yourself is worth the effort. Since this is a hobby project you may think your time is free, but then it has to stay fun as well, hasn't it?
edit
Just got this in a mailing from DesignSpark: fundraising may get you started to have it done by a shop. Erik raised 313 218 dollar for a 5 000 dollar target.
(end of edit)
Alternatively, B: Again, get a quote for a stencil. Yet, even applying the solder paste manually will take less time than hand soldering, which I would not recommend: the resistors and capacitors are not much of a problem, but the ICs may take quite some time if you want to do it proper, i.e. all pins soldered and no short-circuits.
Not A: it takes too long and it's messy. I would only do it myself if I could use the oven.
Remember that Jobs and the Woz also hand-assembled their first batch of Apple computers :-)
Best Answer
Yes, use flux, I've hand soldered QFP's with a soldering iron before.
If there is a solder bridge from reflowing, apply some flux to the bridge, then get an iron and clean up the bridge. Usually the solder will move to either pin and leave the bridge. Excess solder may need to be removed (if there is too much of it) with multiple passes with a tip or with solder wick.
There are a few methods listed here:
http://store.curiousinventor.com/guides/Surface_Mount_Soldering/QFP
In summary of the link above one can drag small amounts of solder across the QFP pins with sufficent flux the pins can be wetted with no bridging. These methods also apply to removing bridges.
Another note:
With flow or reflow, it's better to stop a problem before it starts. Keep things clean and consistent. If reflowing a part, remove as much solder from the pads if your going to apply paste. If your not going to apply paste, it can also be good to remove most solder from the pins and make them look consistent. If you do this the reflowing process will be better.
With any paste soldering if you don't have the right solder and the correct amount, you'll get bridges. If you clean off the pads well, and use a good process then you won't get bridges. You apply the correct amount with the right sized stencil, and must be sized around the pads. Use the recommended stencil pattern and thickness:
Source: https://www.nxp.com/docs/en/application-note/AN1902.pdf