Hi,
What database to use depends on how much data you expect to store (although that is also not that important unless the amount of data is expected to be HUGE). You could save your data in text or XML if you wanted but there would be a reasonable performance hit as the files got longer.
SQL or MySQL should be perfectly adequate for your needs.
To save the data storage problem you could simply request the next question from the database as one is solved, then you're only holding one question in memory at a time. An obvious issue here would be if you wanted the questions to be in random order - you'd need to handle that somehow.
Or you could just extract all of the questions for the quiz and store them in a dataTable in memory (but out of the database) and access them in turn or randomly. This could potentially clog your server depending on the number of questions, concurrent number of users, etc.
I hope that helps,