(a) Is it true it's one of the few airports in the world where aircaft will use full thrust on take-off owing to the short length of the runway?
Not necessarily -- You can de-rate out of GIB if you have got the performance to do it. Usually means either a Flap 15 on the B737 or CONFIG 3 on the A320.
(b) Go-arounds and divertions to AGP are very common in GIB owing to the unusual nature on the winds. Is that correct?
GIB has strict wind limits which mainly affect RWY27 operations and that prevents the 'Have-a-go' types. In my mob you are allowed 2 attempts only then divert. You also need 5000m visibility and a 1000' cloudbase before you can even start an approach.
If you do have to divert you are not allowed to enter Spanish airspace due to political reasons. Therefore you can need to divert either to Tangier (Morocco) or Faro (Portugal). Then you file a flight plan to Malaga (AGP) from there. Then wait for the pax at AGP.(Not too bad now around 2hrs - with the new motorway)
(c) Pilots have to be specially trained to fly into GIB.
Both Captains and F/O's can land at GIB. Capt's are checked out prior to being allowed to operate with a line F/O. Not specific training for F/O's apart from written brief. Capt just need to make sure F/O lands before the road crossing otherwise Go Around !