I don't know about painters tape. Duct tape has been known to rip off scales from snakes, therefore i wouldn't risk any adhesives in the tank.
The probe can go between the uth and bottom of the tub on the outside, then you can check your temps with a IR temp gun to dial them in.
Some people have used hot glue to keep the probe on the inside of the tub but that is not as easily removed for cleaning/moving or anything else.
That is the the correct tub. They measure approximately 14" long 8" wide. It may seem small but that is what you're going for, you want the snake to feel secure. Once it has been feeding well and put on weight you can look into larger housing.
I would not worry about problems from the snake consuming a piece or two of aspen. When I used aspen I fed on it without a problem, not once did a piece stick to a food item ( I don't thaw in water so they weren't wet) but if you are worried you can put down a piece of paper towel or a paper plate as long as you watch they don't swallow the paper towel.
A lot of hatchling racks using 6 qts will use back heat to create a warmer side or by using a narrow strip of heat tape. If you already have a big piece of heat tape I would simply put the tub with only a part of it on the heat tape. (probe between bottom of tub and heat tape)
You should have a small water bowl and a hide. Its hard to tell from the pictures but if your snake can curl up in the hide tightly so its touching them on most if not all sides then it should work fine.
As for sounds I wouldn't worry. Snakes sense vibrations and hear through sound waves so unless you have a blender right next to her she should be fine but to keep stress down it would be best to move her to a low traffic area.
Lastly, the temps should be constant day and night. A night drop is not needed