[Rusty]

Negation in the preterit isn't awkward at all, and is used in everyday conversation.

The 'Cuánto tiempo hace + action clause, introduced by que' structure allows for negation in the preterit, as well. Question 2 is all about that, and it's a good question that can be used when asking about a period of time when an action was NOT taken.

I think what makes it seem awkward to you is that you're using less than accurate English wording.
Translate 'estudiaste' as "did study" (since this is a question, English insists we use an auxilary verb and a main verb).
Translate 'no estudiaste' as 'didn't study'.

1. ¿Cuánto tiempo hace que estudiaste medicina?
- How long ago did you study medicine?
2. ¿Cuánto tiempo hace que no estudiaste medicina?
- How long ago didn't you study medicine?

Question 2 still sounds awkward, doesn't it?
One thing is clear, though. We are asking about NOT studying in the past.

Let's reformulate the translation of the opening clause (prior to the action clause) to see if it sounds any better.

Instead of opening with "How long ago," let's try "How much time has passed."
To introduce the action clause, let's substitute 'since', because we English speakers don't normally say the conjunction 'que' (that) Spanish requires.

"How much time has passed since you didn't study medicine?"
This rendition of the Spanish question sounds a little better. But we can do better.

We could reduce "How much time has passed" to "How long." Then we can dismiss the conjunction altogether, since it isn't a requirement in English.

"How long didn't you study medicine?"
Better?

This reformulated English translation conveys the meaning of Question 2. It doesn't sound awkward at all.

It's clear to see that Question 2 isn't the same as the other three questions.
Each Spanish question was precisely formulated to ask a certain thing.

The only thing that changed in each question was the action's tense.
When negation was added, the opposing action is being asked.

I hope this helps!