[MikeNewYork]

Question #: 1: In spite that no one helped me, I still managed to do it.
User's answer: True
Correct answer: False
Additional Notes: why is it false? Should I say in spite the fact that no one helped me, I still managed to do it?

Either:

Despite no one helping me,
Although no one helped me,



Question #: 4: Although the fact that it was dangerous, he still did it.
User's answer: True
Correct answer: False
Additional Notes: Should I say despite the fact that it was dangerous, he still did it?

Yes, use "despite" there.


Question #: 6: Despite I tried my best, I failed.
User's answer: True
Correct answer: False
Should I say despite the fact that I tried my best, I failed.

You could say that, but I would try to avoid needlessly wordy phrases such as "despite the fact that".

Use "although" here.