Translation:My daughter is with her grandparents.
Yes, „Meine Tochter ist bei ihren Großeltern.“ means “My daughter is [located] at her grandparents['s place]”. But ‘mit’ is used instead of ‘bei’ if she's traveling with her grandparents, as in „Meine Tochter ist mit ihren Großeltern unterwegs.“ = “My daughter is en route with her grandparents.”; or if she's together with her grandparents somewhere other than their place, as in „Meine Tochter ist mit ihren Großeltern im Kino.“ = “My daughter is in the cinema with her grandparents.”.
No. You simply don't use "mit" with persons in these contexts in German.
"Meine Tochter ist bei ihren Großeltern." doesn't state whether she's there right now or permanently. If you want to say that she is living with her grandparents, you'd say "Meine Tochter lebt bei ihren Großeltern." or "Meine Tochter wohnt bei ihren Großeltern."
The difference of "mit" and "bei" is that "mit" is always used if two or more people are doing the same together. So you can "mit jemandem Eis essen gehen" (Going out for ice cream with someone), "mit jemandem ins Kino gehen" (going to the movies with someone), "mit jemanden kochen" (cooking with someone) ect., but if someone is (only) in the care (for children) of another or as a location information - near someone else., you use "bei".
"Meine Tochter ist bei ihren Großeltern." doesn't say anything about what they're doing, whether they're doing something together. It is meant to say that you don't have to worry because the grandparents make sure that nothing happens to her (or that she doesn't harm anything/anyone else) OR that she is at her grandparents' house or with her grandparents somewhere else. (So you know where to pick her up or that you can call her by calling her grandparents' (cell) phone number.)