My cousin was on a respirator and then had a trachea put in. How long before she can breathe on her own?

Weaning a patient from a respirator (mechanical ventilation) can vary, depending on the exact combination of problems. It could takes weeks, some patients are never be weaned.

The underlying problems that caused the breathing failure, chronic respiratory problems she already had before the illness, problems with nutrition during the hospitalization, new infections, general muscle weakness, and a ton of other problems can will all have to be addressed. The longer she has been on the ventilator, the more complicated it gets. The staff there will be your best source of input.

If she is on a ventilator for pneumonia and a trach (tracheostomy) was necessary, it would indicate she has been on the ventilator for a while. The decision to do the trach may mean they think she will be on the ventilator for a long time, or that the trach would make weaning easier

Only the family can decide on the issues surrounding her son's death. If she is not "awake" she may not be able to understand, but it would surprise you what patients ARE able to process even if appearing confused or disoriented.

Your cousin is still in very serious condition. The respirator and the tracheal tube are there to make sure she does not suddenly stop breathing even if she can breathe on her own.
It is highly unlikely that she will ever be well enough to attend her son's funeral. At this point, I don't think you should tell her that her son had passed away, as this would have been too traumatic for someone in her condition. Wait till she is well enough to sit up and off the respirator before telling her. Depending on how badly she was hurt, it may be weeks or months before she can breathe on her own.