Welcome to the board. Your DD sounds wonderful. I loved her logic about lunch location. haha

Is she gifted?

JMO but she definitely sounds like she is. My DD will be 4 this next month and she was very much like your DD as a baby. First word right before she turned 3 months: elephant but very verbal before that and technically said 'hi' at 2 weeks but I still don't count that one as her first word. Talked in complete sentences by 6 months and crazy long complicated sentences by 15 months. She always understood pronouns and used them correctly from the start. She also always had an amazing attention span and such imagination. She, too, was all about someone reading to her and took it to the extreme. If given the choice of a toy or a book she always wanted a book.

Do I absolutely know if my DD is gifted? No and when I first started to figure it all out I quickly jumped to we need to have her tested but came to realize that it was unnecessary because who was the test for? It was more as documentation to prove it was true when I don't really need that. I know she is in the highly gifted range just as I know she is a blond hair, blue eyed little girl. When and only when I need to test her for school or because she might be having emotional issues or whatever will we test her. But until then I will just accept her for who she is and enjoy her.

Enrolling her in school:

I was in the same boat as you are now. Shouldn't she be in a preschool to help socialize her was my question and I raced to find one but all of the preschools in our area had a waiting list so DD didn't go to school while she was 2. I don't think it hurt her one bit. We did other activities such as dance and she got to know some little girls that way. When she finally did go to school she was standoffish but with time she came around and gets along with her classmates.

If you feel that she isn't getting enough socializing then maybe you consider it but you might just find some hour activities through the week that works for you too.

As for teaching her to read; just be careful with this one. She might be like my DD and resist any attempt. My DD started reading right before age 2 which made me think, okay she is ready to learn to read. (And by started to read, I mean word recognition in books and signs) But the minute I sat down with her to 'teach' her, we hit a big wall and she out and out refused and we spent the next year pretending like we didn't notice that she just read something. Then a few months back she came to me showing me she could read and asking me to sit down with her and teach her some more. I still wonder if we didn't make such a big deal about her abilities back when she was 2 if she would have been reading a long time ago. She had all of the tools since before 18 months.