A USB drive with important data is not a "first time soldering project" repair. You stand a good chance of doing more damage, especially if you don't have any good soldering tools.
For board/connector level repair like this, you're looking at (ideally) just putting a new connection on, but perhaps dealing with some board level repair as well if the traces have cracked. It's doable, but it's not a "Hey, let's learn to solder!" project.
In extreme cases, you might actually need to transfer the components to another board - it depends on what the damage is and how easy it is to work around the damage.
There's no reason to bother with cold solder - you can safely solder components with the proper equipment and temperature regulation. Pretty much all spec sheets have a "safe operating temperature" range and a "soldering temperature range" that covers how you can heat it up for soldering. You don't want to operate it there (also, you'll melt it off the board), but they're not particularly fragile.
All that said, if you don't have a low level expert locally who can do surface mount soldering and work, I'd be happy to take a stab at it. I can't guarantee anything without seeing the hardware, but if it's relatively straightforward damage, I can work around it and should be able to get the data off. I don't have a discrete chip reader or anything, so I'd be using the existing board if it's repairable.
Do you have a way to disassemble the case and get high res macro shots of the damaged areas?
... also, sol, no, most USB drives are not uSD cards in a reader. I've been inside quite a few, and they're almost universally surface mount hardware - possibly two separate dies for the controller and the memory, sometimes a single combined die.