Actually, you have done your daughter a great favor. To address your specific concerns:
She's not likely to be considered the weird Chinese girl because she does speak English, usually the problems come when the child only speaks a language different from the kids. In fact most kids think it's pretty cool when another child can speak a second language.
As far as the speech problems, studies have shown that learning a second language as a young child has absolutely no negative effects on speech. However there are many positive effects of teaching a child a second language at a young age. First, their brain actually wires the way it deals with languages differently, this makes learning future languages much easier for these children, and also makes learning more complicated concepts in the languages they already know a little easier. Also, you are showing your daughter that you have a respect for her heritage, and are giving her a connection with that, which will increase her self-esteem.
For many years people have doubted whether it is wise to teach a second language like that, as there are many old wives tales about it. But study after study has shown that it is very beneficial. The only negative that they have ever found is that sometimes they are a tiny bit delayed in their first words as they are absorbing so much more language, but once they start speaking, they actually attain language at a faster rate. Since you are past that stage anyways, you should be in great shape.
I commend you on putting the effort into getting her someone to teach her Mandarin, and strongly encourage you to continue.
It is possible that they are jealous, but more likely they are just uninformed. A lot of people really do believe that it can cause problems, however those who have researched it and studied it have shown just the opposite, that it strengthens the child. I know it is hard when you are constantly faced with negative comments, the best bet is to learn the research so that you can respond to them, and then just know that you know what is best for your daughter.
Here are a couple of articles that might help you: