No matter the bulb they get hot. Any bulb in the cage should be inside of a proper wire barrier.

Plastic sliding doors are going to scratch easily. Also large ones will flex and may not be as secure as you need. Glass would be better and IMHO is not very excited to get cut.

The main issue with wood enclosures is the high humidity causing issues with it(bubbling, mold, etc) and being hard to thoroughly clean.