When I put down a fluid cow from a safari net, nothing comes out of it and my safari net becomes empty. T_T lost a good fluid due to that :/
The mob limit is for a 3x3 chunk area. So make sure you are at least 3 chunks away(4 would be better), then try it. If it spawns, then you have too many animals. If not, then please let us know
I don't know but in my experience sometimes the mob is just invisible. If I were you I would use golden lasso (although it can only keep passive mobs), it never fails even when I have a mob farm turned on near me.