Well as you are asking...
While a teacher may sound good... I don't think they would have enough experience of caring for a baby. I do not feel that the agency has matched you with someone who has recent experience of caring for a similar aged child.
The teacher may not have: First Aid Training, especially Infant Resuscitation. Now they may well have that training... as a nanny I have that training updated every 3 years... are all teachers first aid trained?
When in strange places, children can sleep differently. Keeping your DD to your usual evening routine should help. How your DD reacts to someone else putting her to bed though will be an unknown... with luck the teacher will have the patience of a saint and be happy to sit with DD for as long as it takes (if that is what is needed to keep DD calm).
With first timers, I use TXT messaging to keep in touch with the parents (well usually it's the mother who wants the updates). I prefer to send the occasional txt to keep mum informed, rather than having mum phone ever 30 mins.
Many parents will use a babysitter they have never met. Many parents say it goes fine.
I'm not a parent so I don't know what it is like to leave a child with someone I've never met... but my gut feeling about it is that I would want to meet the person first. While the agency are sending the person an hour early, what happens if you don't feel the person is suitable - I guess you still pay and don't go out.
For a relaxed evening out you need to have as much confidence in the person who is caring for your DD as possible. With luck the teacher will bring with her a portfolio of childcare references, training certificates, Criminal Record Check, that kind of thing... so you can have a look through it and give yourself the confidence to leave your DD in their capable hands. As they are coming an hour early... your DD will also have some time to get to know them.
At the moment, I think option C.
Why the agency could not have assigned you a Nanny, Maternity Nurse, Nursery Nurse or Childminder I don't know... surely any one of those would be a better match than a teacher (who may have lots of experience with 5-10 year olds, but not babies).
Yep, still "single and looking"
I occasionally chat with a lady my-age-ish in New Zealand... wonder if your friend is the same person - doesn't run a childcare company does she?