As others have said, bulbs in the cage are a no-no, unless protected by a wire cage like pictured above. Cords shouldn't be loose or climbable either, they can actually act almost like a noose.
You're right, no tape... I know of many keepers who use a hot glue gun to glue the probes, etc...in place.
As for the burn, it depends how bad it is. I'm not personally experienced with burns (on snakes, I cook for a living so know all about burns on myself, hahaha)...
...but maybepost a few pics so more experienced keepers can help you out.
Good luck!