How do you write #1? Does #2 need a comma before the word, while or no comma at all?
- "How is it? Do you like it? she would ask.
- And that is how we would carry on with our mother/daughter dinners each night of the week while my dad was at work.
The first sentence is correct if you add a quotation mark after the second question (see Handling quotes ending in exclamation or question marks and this guidetogrammar page):
"How is it? Do you like it?" she would ask.
The second sentence is a case of joining an independent clause (everything before the "while") with a dependent one ("while my dad was at work"). Because the independent clause comes first here, you do not need a comma (see butte.edu tip sheet).
And that is how we would carry on with our mother-daughter dinners each night of the week while my dad was at work.