Translation:I do not like sports very much.
They are part of a bigger word, ではありません ("dewa arimasen", which is a formal negative form of the verb です, meaning to be. So, the ではありません together really just means "is not" or "do not." I hope this helps :)
ではありません is the negative form of です. They ate not used as particles here.
While I think I understand what you are trying to say, the phrase 'I don't like many sports' is somewhat ambiguous. It could mean that you dislike many sports or that you have a preference for only a few sports.
Assuming the first reading, the Japanese would be 「きらいなスポーツはたくさんある。」(There are many sports that (I) hate).
Assuming the second reading, in Japanese it would be 「好きなスポーツは少ないです。」(Sports that (I) like are few.).
While あまり or あんまり can be used as a noun or adjectival-noun, as regards the usage of あまり in the sentence example it is an adverb.
With respect, 'amari' is qualifying the word 'like', not the word 'sport'. I.e. Sport in general - 'not' liking 'much'.
'I really don't like sports.' means something different from 'I don't really like sports.'.
Same meaning. Different level of formality.
Both are used, so both should be learned.