A BGA rework professional may be able to swap the memory to a new flashdrive, but without a hot air station you don't have a hope of switching it. Try pressing on the chips while you plug it in. Maybe it will work long enough to get your data. If not I would have to say send it in and pay. :(
I agree mostly with old turkey. But If you just connect the top two spots together (not the two on the right) you should get the ear speaker working again. Not the way it should, but working. If you can, you should get the right parts. What oldturkey didn't tell you, is that the missing components in your picture aren't for the camera, so you need to hunt for something else to fix that. You may want to trust the advice of a more experienced person than me though. eg. oldturkey ;)
Your iPhone has two lines for the backlight. If you had water damage, one of the lines is probably down causing half of the screen to be darker. You will need board repair. Probably a backlight filter. There probably won't be a whole lot of shops willing to work on it, because of the water damage.
There could be a light on that you aren't noticing. There could also be a short in the wiring some place. To narrow down the problem you would put a multimeter set to the highest amperage it has in series with the battery and remove the main fuses one at a time until the current drops to approximately 0.2 A or less. That's the circuit the short is on. This should be done with the key off and the doors closed. The owner's manual should describe what each fuse is for. Is there an under hood light on that vehicle? It could be bent into a position that leaves it on.
As DC Life Tech said, you can switch the logic boards provided they are the same model ie: both iPhone 5 or both iPhone 5S. It wouldn't do what you are hoping though. The memory is permanently attached to the old board. The new board would have the new memory with it. Hopefully that makes sense.
The component with the most damage is the capacitor that is burnt there. It goes from vcc main to ground. If it is shorted, which is very likely judging by the looks of it, you will have no power to the rest of the phone. I would test it for a short and remove it (WITH a soldering iron) and test for a short to ground again. If it is not shorted anymore, start the phone making sure nothing gets to hot to touch. The chip there is the wifi/ bluetooth chip. If you knock any of the other components off the board though, you may lose wifi and/or bluetooth. Definitely clean and dry the board too.
While in a call, press the volume up button. It will show the volume on the screen and adjust it, just like with the normal loud speaker. If that doesn't work, try taking it apart again and cleanup the connections for the earpiece speaker, including; the contacts under the speaker, and the flex ribbon cable on the logic board.