No, neither one is correct English. The first one doesn't mean anything, though the second one can be understood. However, none of the mistakes have to do with none/nevertheless, which is used correctly as an adverb in both examples.
The answer to the question
Are these words used as a replacement of the phrase in spite of?
is No. They mean roughly the same thing as that phrase, but they are not replacements for it.
Nonetheless and nevertheless (which mean the same, incidentally) are always adverbs,
and may appear in many adverb niches:
- Nevertheless, he passed the exam. (comma optional)
- He nevertheless passed the exam.
- He passed the exam nevertheless.
Whereas in spite of (equivalent to despite, which falutes slightly higher) is a preposition,
and must appear with a noun phrase object.
- He passed the exam, despite/in spite of his alarm clock.
- He passed the exam, despite/in spite of the weather.
- He passed the exam, despite/in spite of being malnourished.
The entire prepositional phrase itself (including the object noun phrase) is a sentence adverbial,
and so it can be niched like an adverb
- Despite/In spite of the weather, he passed the exam.
But in spite of by itself can't be used as an adverb:
- *In spite of, he passed the exam. (ungrammatical with or without comma)
- *He in spite of passed the exam.
- *He passed the exam in spite of.