I have a really random question. When somebody starts using Ask Ubuntu, they start with 1 reputation. According to the "privileges" page, you must have at least 1 reputation to create a post. If, for instance, a new user creates his/her first post, and it gets a downvote, would they loose the right to post questions?
No, the minimum rep is and will always be 1 no matter what, that is the minimum necessary to ask a question and will be the minimum for a user after reputation recalculation.
If a user drops bellow 1 the system will compensate for that and after a recalculation it will be back to 1.
There are several reasons for that already discussed in Stack Exchange meta, have a look:
Not how you mentioned above, however moderators can impose a question ban, or can suspend you so you can answer questions/edit etc, but can't ask questions.