Does Closed Septoplasty Weaken the Nose?

No.  If performed correctly a septoplasty performed via the external or closed approach should not weaken your nose.  When performing a septoplasty an adequate strut needs to remain intact particularly in key areas of the nose.  Septal collapse is a known risk and complication of septoplasty but this is mostly secondary to overaggressive removal of the cartilage and not leaving an adequate strut.

Most surgeons performing septoplasty today adhere to several principles that will maintain the structural integrity of the septum and the nose.  Typically deviated portions of the septum are removed during the procedure (bone and cartilage).  This must be done judiciously to prevent collapse of the bridge (dorsum) and cause something we call a "saddle nose" deformity where the bridge of the nose collapses and causes it to look like a saddle.  Your nose will remain strong as long as sufficient bone and cartilage is preserved.  

Your question about whether or not you should avoid contact sports is a bit more difficult to answer.  If you look at a lot of boxers, hockey players and MMA fighters that have repeatedly had broken noses, many of them have "saddle noses" without ever having had a septoplasty.  Septoplasty will not significantly increase the chance of your nose collapsing with contact sports if done properly.   

A septoplasty should not weaken the nose because there is always 1-1.5 cm of both dorsal and caudal cartilage leftover in the nose.  The cartilage and bone that is removed is typically nonstructural so it does not weaken the nose.  It is best to have any septoplasty procedure after a patient is done with hard contact sports since patients do not want to reinjure the work that has already been done.