I want to use this radio module and this antenna and this extender for my diy RC. Is this possible, and how would I connect the extender to my radio module? Can I cut the U.FL side and solder them directly to the board?
Yes, you could do that, but it will be challenging. You have to remove the outer jacket, separate the shield, strip the inner insulation, and solder shield to the gnd pin, and center to the ant pin, on the radio module. The slight glitch in impedance will not be an issue, but it will be a fragile connection mechanically and may fail from shocks and vibration.
The connection points on the radio module aren't suitable for soldering a U.FL receptacle. But I think you could instead place an edge-mount SMA receptacle. This appears to also be available from the same company that makes your radio module (here). Then just use a suitable SMA to reverse-SMA cable (also available from same source).