Definitely put the mat on the bottom -- as Kaorte said, as long as you have a working thermostat attached, your snake won't be burned.
When I first set my tub up I was paranoid about this as well. I finally realized that pretty much everyone in the herp world uses tubs for their snake. They have to heat them somehow, and if it were dangerous, folks wouldn't keep doing it and recommending it. Even at full blast, that UTH should not get above more than 140f, and that is not nearly enough to even melt the plastic, much less burn it. Just make sure you follow all the safety guidelines that came with the UTH -- ie, throw away if kinked or bent, & allow airflow.
As for feeding in the tub, I have not really heard or read anything about many cases of cage aggression with ball pythons. I would say that at the very least, you should feed in the tub she lives in until she is an established eater.
Good luck with the BP, you will have a blast
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