I'm rather positive this is not the "approved" method to update a survey, but at the moment I'm just trying to get this moving....

I have a survey I want to update.
If I deactivate the survey, add a few questions, then reactivate the survey, will the survey continue to work as before?
That is, will is maintain all the previous response data?

Additionally, to save the current data, with export option do I choose?
Will any of the methods save all the questions and response data?

Data and response data will be saved, but will be archived (moved to another database table), if you deactivate the survey. If you are lucky (which depends on the changes made to the survey) you will be able to reimport those older responses to your survey again, when you activate it. LS will generally ask you about that. However, I don't think there is a guarantee.

Whenever you deactivate the survey, you should export the survey responses before. At least they are safe and you have them. If everything goes wrong, you can later join the two different tables manually in Excel. Usually this isn't a lot of work, you just need to find the new columns that you have inserted (or which ones you have deleted) and then adapt the two files accordingly. Generally a quick fix, if you don't make too many changes.

Of course, if you are not able to reimport the old responses into the system, you won't see them in the report/statistics.