In statements you say "sie spielt" and in questions you say "spielt sie". In subordinate clauses you say "sie spielt". Note that in statements it can be in any order as long as the verb comes second. E.g. "Sie spielt mit dem Hund" = "Dem Hund spielt sie". You can judge which is the object by the declension of the definite article (dem)
In the dative, feminine "die" becomes "der." (I wouldn't say it becomes masculine, it happens to use the same article.) So "meine" becomes "meiner," "eine" becomes "einer," etc.
Die Tochter spielt. (The daughter plays. Nominative.) Der Hund spielt mit der Tochter. (The dog plays with the daughter. Dative.) Ich kaufe der Tochter den Apfel. (I buy the daughter the book. Dative, then accusative.)
"Tochter" becomes dative when she is the indirect object. Also, whenever you see certain prepositions like "außer," "mit," "von," "zu," "bei," "seit," and "nach," it's dative.
I'ts a different tense.
Spielt sie mit meiner Tochter? = Does she play with my daughter? / Is she playing with my daughter?
Spielte sie mit meiner Tochter? = Did she play with my daughter? / Was she playing with my daughter?
The object of a preposition can be either accusative, dative, or genitive, depending on the preposition. For example, "durch" always needs to have an accusative object ("durch den Tor"), and "von" always takes the dative ("von dem Tor"). "Mit" happens to take the dative.
See this link for help on preposition cases: http://germanforenglishspeakers.com/prepositions/introduction-to-prepositions
Because "Tochter" is feminine and is in the dative case here (because it follows the preposition "mit"). As you can see from the second chart in this article, the correct ending for feminine dative is "-er."
"Meine" would be used if "Tochter" were being used in the nominative ("Meine Tochter spielt") or accusative ("Sie sieht meine Tochter"); "mein" is only for masculine and neuter nouns and so would never appear with "Tochter."