Welcome to the site! Hopefully I can answer most of your questions...

1. Stuck pieces of shed are usually not a problem as they will normally come off during the next shed cycle, however... if the shed is stuck on their eyes or around their tail you should remove it. Eyecaps can lead to permanent eye damage and tail sheds can lead to reduced circulation and eventually necrosis of the tail. Fortunately its pretty easy to remove stuck shed pieces by soaking your snake for 10-20 minutes in a lukewarm bath and gently rubbing the skin to remove it.

2. Most of the "problem" woods are aromatic varieties like pine and cedar. Hardwoods should be fine and an easy method to disinfect it is to place it in the oven for 20 minutes at 200F. This way you don't have to worry about harsh chemicals like bleach soaking into the wood and coming into contact with your snake. Also, try to get a smooth barked wood like manzanita as it will be much easier to rinse off and keep clean if it becomes soiled.

3. Yes, you should be concerned about the exposed light in your cage. Many burns occur because snakes have access to exposed bulbs or heat emitters. A simple cage can usually be constructed or purchased that will prevent your snake from coming in direct contact with the bulb.

Good luck, and feel free to ask any other questions that come to mind!