Can I use 2x4s to extend studs so I can mount my TV?
This should be OK, but I would recommend attaching the 2x4s to studs on both sides of the TV, not just the right side as you've shown. You don't want the weight of the TV to pull the 2x4s away from the wall on the left side. A 55" TV is going to be heavy and the eccentricity caused by the spacing of the wood and the bracket will cause it to pull away.

Personally I would probably use a single piece of plywood — it will have a lower profile and will provide plenty of opportunities for getting strong fasteners into the studs. But 2x4s will be cheap (or free, if you've got some lying around), are easy to cut, and should work fine.