I have no idea how "transponder" comes into this, but what you are describing is specs on a satellite antenna.
Left hand or Right hand circular polarization (switchable)
Gain 11 dBi I will assume. (gain relative to isotropic)
Frequency 402.75 MHz +/- 200 KHz (no problem there)
3dB beamwidth 40 degrees... Is this a minimum beamwidth or max? It is also tied in with the gain figure.
VSWR 1.2-1 (I assume this is over the 400 KHz bandwidth).
For connectors, I would use type "N".
If I were designing it, I would use two yagi antennas on the same boom. One vertical polarized and one horizontally polarized. They are fed 90 degrees out of phase to attain circular polarization, and also to attain switchable left or right handed circular polarization.
Assuming you know nothing, Yagi's are like TV antennas.
Size for such a critter would be approximately 1.3 meters long, and 35 cm wide.
Wind loading would depend on the design of the final antenna.
Basically, if this is the info you want, forget transponder, and look at antennas.
The American "Amateur Radio Relay League" has an antenna book which gets into sat antennas You might find it helpful.
Sincerely
Bill
ps what you do when the signal gets to the ground is up to you (AKA transponder).