Any uth should hook up to a tstat. They are a must for uth. Using them without is very risky. I would not move the uth to the bottom until you get a tstat. When you do remove it do it slow and easy. Bending the uth to much will cause the element inside bend and break causing the uth to malfunction. Once its removed you can put foil on the sticky part and then tape the uth to the bottom of the tank. Doing this will allow you to remove the uth whenever needed in the future.
As for you current tank I would cover the majority of the screen with something. I use foil wrapped cardboard. This will help hold heat and humidity.
Also the branch and branch that you are using as a hide is not a very good option. Also the half log is better than the branch but still not the best for a snake. Ideally you want a hide that has one small door that is snug to the snake. The log is offering some security but the branch is not offering anything.
Placing the log against the back wall is good and you could make it even better by adding moss to the front to create a door of sorts. Moss is also good to make a hide that is to large feel smaller and more secure to a snake.
These are great hides because they are sturdy, cheap, and easy to clean.
http://www.reptilebasics.com/hide-boxes