- Niobe also had seven sons.
- Niobe quoque septem filios habebat.
quoque follows the word which is to be added/emphasized, so it should come after filios (she had seven daughters and seven sons also).
quod simply means “because” here so it is fine.
- Latona had only one son and one daughter.
- Latona unum filium et unam filiam solum habebat.
solus, -a, -um is modifying both filium and filiam, so it would usually be accusative plural (masculine, since there is a male in the group).
- Latona was irate because Niobe refused to give gifts to Phoebus and Diana.
- Latona irata erat quod Niobe Phoebo et Dianae dona dare negabat.
Correct, but I think the perfect might be better to use.
I don’t think this is “refuse” in the sense that is required. I don’t know what the proper word would be, unfortunately; if I had to do this myself I would change the wording to “…was unwilling to give gifts…”. I don’t think the perfect would be any better than the imperfect here, but it is unlikely that the student knows that tense judging by the content of the exercises, Dingbats 
- Niobe said, “Women of Thebes, worship me; give gifts to me, not to Latona.”
- Niobe inquit, “Orate me, Feminae Thebarum; date mihi dona, non Latonae.”
Instead of feminae Thebarum, you could just use an adjective to say (feminae) Thebanae. When inquit is used, it usually breaks apart the quotation, as:
“Orate me, Thebanae feminae;” inquit Niobe, “mihi date dona, non Latonae.”
Notice the emphasis is on mihi by putting it first, creating an emphatic contrast between mihi and Latonae, much like English italics: “give me gifts, not Latona”.
- Niobe said, “Fortune is good to me.”
- Niobe inquit, “Fortuna bona est mihi.”
Again, inquit usually goes inside the direct speech (quotations), e.g.:
“Fortuna,” inquit Niobe, “bona est mihi.”
- Behold, Niobe is queen in the palace of Cadmus.
- Ecce!, Niobe est regina regia Cadmi.
I’d put > in regia > or > regiae> .
I agree, the dative would be a nice touch, though the preposition + abl would also work. “of Cadmus” could also be rendered by the adjective Cadmeus, -a, -um, though the noun Cadmea, -ae by itself refers specifically to the citadel at Thebes of which you are speaking.