You’re not crazy or unreasonable. 
You’re living with this uncertainty every day, without any other family/friend support the only other person you can talk to is your DH, and he dismisses everything you say, no wonder you’re constantly doubting yourself. This must be so hard and you must feel extremely lonely so I can totally understand why, this job and the probationary period is extremely important to you, it’s where you feel in control the most.
With this in mind and to help you manage your overwhelming feelings, why don’t you think to the future in short bursts.
Think of the current nanny now as being a short term solution, rather than being with you forever. Start to make enquiries now on other agencies, notice period for your current nanny, etc. Once your probationary period is over (how much longer do you have?) or even before, could you maybe interview one nanny each weekend, so as to not feel overwhelmed by this? Therefore giving yourself lots of time to organising a replacement. I wouldn’t speak to your DH about this as he dismisses your feelings at every turn.
Regarding current nanny and the loans, advance in salary, staying at weekends etc. If you think you’re not going to get that loan money back then suggest that during weekdays she could work through a couple of nights if your DD needs settling, rather than you getting up. Tell her you’ll deduct what she owes from the loan each wk.
Tell her you can’t advance her anymore as there is already the loan outstanding and hope she understands. Ask if she’d prefer to be paid weekly? Give her that option if you feel uncomfortable telling her no when she brings up money issues etc.
The weekend issue when she arrives unannounced into your home when she’s only contracted each wk. Again, you’ve let this go a few times so probably now feel like you can’t now bring it up but you need to address this. As another poster suggested, you could leave your key in the door, double lock it or if you feel able, bring it up with her.
I think if you can start making small steps to where you feel in control, no matter how small, then you will start to feel more confident about you as a person and a mother. It’s understandable you doubt yourself at every turn, as your DH dismisses everything you bring up, you have no one else to confide in so no wonder you wonder whether it is you rather than the nanny.
I’ve read the entire thread and imo you are right to feel the way you do.
Regarding the way your DD has bonded with your nanny. She will have this bond again. I’m in the always felt sad when DS moved rooms in the nursery and would have a different key worker the following year. Sometimes it took him a few weeks to settle, then he never looked back. I know it’s a different scenario but our dc can be more resilient than we give them credit for and it’s often us that worries more about change.
As you’re in the US, can you share which area you’re from? There must be other US Mumsnetters who can help regarding the law around notice period, maybe helping to search for other nanny agencies etc
I wish you well, OP and hope this all falls in to place for you. Remember, take little steps so that you don’t feel overwhelmed by any decisions you’ll need to make.