RE: reciever antenna solder or replace?
Soldering.....the joint should be much less than one ohm, and even if the antenna is matched for 50 ohms, the joint is neglible. As said above, repaired breaks don't fit through antenna tubes very easily, although for your break, that's not a problem. I'd use 22 gauge stranded wire (for strength) and replace the whole thing anyway. Do a ground check with the transmitter antenna down and make sure you still have control at around 100 feet after that.
It doesn't appear that antenna length is based on wavelength all that much, as regardless of frequency, antenna length always seems the same. I thinkk they just pick a generic length and then tune the receiver input matching circuit for best reception.
To add further confusion, .....from the Futaba website:
approximately 19 1/2" for surface (1/2 meter)
approximately 39 1/2" for Air (1 meter)
Although the length makes the antenna efficient, it's the amount of antenna standing straight up that counts the most, and the more standing up, the more strength the signal will be at the receiver. But it's not that critical. Our little cars/trucks arn't driven very far away from us. For people wanting the best, get as long an antenna tube as you can...the more receiver wire standing up, the better.
Keep your transmitter antenna upright as well to maintain vertical polarization and therefore, best range.