Can I use a semicolon when linking a sentence fragment and a question?

No, none of those semicolons work. A semicolon is used to connect two independent clauses (i.e. things that could be a complete sentence) that are related to each other. If you have only a sentence fragment on one side, a semicolon cannot be used; normally a comma or a dash would be called for there.

In several cases, you have a topic fragment and then a specifier fragment (e.g. "You bought icing; chocolate, right?"). In those cases, a colon is correct.

A colon can also be used to introduce a list.

(It's worth bearing in mind, given your note that these are transcribed conversations, that people do not always speak grammatically or in complete sentences. However, in all of these cases, I think you can come up with something reasonable.)

I would go with the following:

I was trying to leave, asking them to let me go.
She kept texting me: "What's up," "Want to go for a walk in the park?"
How did you feel about that, not having to go to work?
So why did you say what you did: "No," and later "I don't know"?
So about that cake—you made it yourself?
Talk about that for a moment—your tardiness, that is.
You bought icing: chocolate, right?
Is that something you regularly do, buy chocolate icing?

see This "The Oatmeal" cartoon for good information about how and when to use the semicolon.

Semicolons are great for lists that include some complex items, ones which might not be clear if commas were used. The items need not be clauses. Parallelism has to be respected amongst the items.

In other words, yes, you can use semicolons: put all eight corrected items in a list separated by the semicolons.