Your discomfort doesn’t appear to stem from any peril to your daughter, but rather from your own biases. Because our transphobic society has told most of us that trans people are in a special category, that’s why. The best thing you can do for your daughter is to wrap your mind around that. You aren’t alone in feeling afraid and uncertain at various points along the way as you watch your daughter explore things that are foreign to you.
My older Latina mother, who lives with us, disapproves. She goes to a small private school where she would be labeled by some, although there are friends who would understand.
This most likely will hurt your chances of other girls wanting to be your friend or date you. Look beyond right now; consider yourself in 10 or 15 years. It’s good to have friend relationships with girls outside of a dating relationship.
That’s more time for you to learn about girls and what your preferences are without being in a relationship. Call them to talk or ask them out (Ask them out well in advance of the date night).
Take responsibility for what to do and come up with ideas based on what you think she would like.
Suggest a plan, but be flexible if she has some different ideas.