Are aspergers people aware and bothered by the fact that they lack social skills and nay be socially awkward? Or do they not notice?

Social awkwardness. Most people with asperger's disorder want friends and to be able to join social activities. They lack social pragmatic skills. Social pragmatics can be taught. .
Individuals differ. Asperger's syndrome has very wide variations - even to the extent that it is controversial. So, there may not be an answer to your question. Also, in addition the that persons level of social skills and other faculties, the severity of the others' reactions would matter. Few would be so unaware as to not notice cruelty.