If the reverse syncronizer is bad could that cause reverse to slip out of gear and grind?

Also, My first gear grinds while I take of from a stop, it did not do this when i got the car and we know it is not the clutch, it stays in gear and the grinding goes away after accelerating. Do you feel this could be a bad gear? Please post any suggestions.

 

What car & transmission do you have? Does first gear grind when you are trying to shift into it. or does it grind while the car is moving? Your clutch may not be relasing fully causing you to grind gears. Otherwise you have a problem. Jumping out of reverse is kinda odd. It could br the synchro, or possibly the shifting mechanism.