From Punctuation in Dialogue (source: The Editor's Blog)
Dialogue interrupted by dialogue tag
Dialogue can be interrupted by a tag and then resume in the same sentence. Commas go inside the first set of quotation marks and after the dialogue tag (or action).
“He loved you,” she said, “but you didn’t care.”
“He loved you,” she said, hoping to provoke a reaction, “but you didn’t care.”
Separating this into two sentences also works. The first sentence will end with a period and the second will begin with a capital letter.
“He loved you,” she said, hoping to provoke a reaction. “But you didn’t care.”
Note that "separating into two sentences" only works if the dialog tag is inserted at a point where the quote can logically make two sentences. You can't use a period with something like this:
"That is, without a doubt," he said, "the worst pirate I have ever seen." -ok
"That is, without a doubt," he said. "The worst pirate I have ever seen." -wrong
So, your initial sentence cannot stand without either a comma or a period, but either of your suggestions can work (although if you use the comma, the "and" of the continuing dialogue should be lowercased); it just depends on how you want it to read.
(The way I interpret it, the comma makes it read more as two simultaneous questions that occurred to the speaker, while the period makes it read as one main question that was already in mind, and then a follow-up.)