Numbat.

Correct. Said to be = sounds like, unfeeling = numb, vampire = bat.

Unsympathetic locals confused at hearing you? (1 word, 7 letters)

Exactly.

 

Unsympathetic = whole world clue

Locals confused = rearrange the letters in locals

hearing you = word or letter that sounds like you = u

localsu scrambled to make callous

 

 

Sounds like sheep and golf accessories skirt the issue (1 word - 4 letters)

I think I'm wrong but,

 

Bate?

no, sorry