Im struggling to trust that she is a good person.
Why? I appreciate you might not like her because your ex husband left you for her, but has she done anything to suggest she would be untrustworthy around a child? Very different things.
Who isn’t going to either run off back down south with my baby
You think she's going to kidnap your child? How exactly are you imagining this running off with your child? This is preposterous.
or hurt her in some way.
OP. Please get a grip. Unless there's a massive drip feed coming that the new woman is a drug addict/alcoholic/released from her 8th stint in prison, there is nothing at all to suggest this. You sound like hard work. Making completely irrational and unfounded comments as if by coming out with this nonsense it's only acting in your child's interests.
For my child’s bedroom they purchased a double bed. In the thoughts that when this woman’s friends and family visit they can sleep in the bed that my child would sleep in if/when she stays over.
If they don't have enough room for a child's room and a spare room, this is pretty standard practice. They will change the sheets obviously. It's no different to you staying at a friend's house and them letting your DD sleep in their spare bedroom. You'd be ok with that. It's the same. It makes perfect sense if DD is only going to be there for example, EOW, that they have a room that can be used more than 4 days a month. It's for any friends/relatives. Not just "that woman's."
You need to separate your hatred for her, from a father's relationship with his daughter.