Your question is a bit too broad for Stack Overflow, but here are a few guidelines.
Yes, all you said is possible! You can start from something as simple as:
['How many moons does Earth have?', 1, 0, 2, 3]
knowing that the first answer is correct and shuffling them for your presentation.
You can even make a separate button for each answer (improving the UX by getting rid of a redundant click) but beware of accidental clicks in that case. It's all up to you.
Here is a super-simple quiz I made for you: http://jsfiddle.net/rTv97/ There are millions of ways to improve it, of course. But it should get you started.
Note the system I used:
['How many moons does the saturn have?', 1, 31, 35, 29, 48]
\ / ^ \ /
text correct answers
It goes: question text, correct answer index (one-based), answers. So, number 1 here means the the first of the 4 answers (31, 35, 29, 48) is correct.