Can animals make their own unsaturated fatty acids?
Yes, animals can make their own unsaturated fatty acids. 

Mammalian fatty acyl desaturases can introduce double bonds at the &Delta;5, &Delta;6 and &Delta;9 positions (i.e. numbering from the functional group). As shown in the diagram below, this means that we cannot introduce double bonds at the &omega;3 or &omega;6 positions (i.e. numbering from the methyl end of the molecule) in fatty acids of a reasonable length (see any Biochemistry textbook for confirmation). 

I've illustrated the impossibility of us making &alpha;-linolenic acid as an example.