There is a lot I don't notice but other animals I do. You're right, many people don't. When I walk around I'm aware of the birds that cross my path and stuff. But sometimes I see birds scrambling to get out of the way for other people because the people don't notice them and don't think that they even deserve their notice.
I'm torn on the insect issue. I don't want to kill them. However I'm deathly scared of a few types. I usually get someone else to help me with those. Remove them I mean. If possible.
Honestly, the best advice for dealing with scary insects/animals is just to learn more about them. I can think of more than a few bugs that look horrifying, but really arent. Ever seen a house centipede? If you want to see something that looked like it crawled right out of a nightmare, go look it up. Alot of people REALLY hate those things. The fact that house centipedes are *very* fast and quite large doesnt help... they could be on the ground one second, and in your hair the next.
But... they're totally harmless. In fact, they hunt and eat other insects that could be genuinely classified as problematic pests (ants, termites, and so on). My mom's house, there's alot of house centipedes for whatever reasons. They are very large. But I just ignore them when I'm over there... because I understand that they arent even remotely dangerous. Granted I can get a bit startled if I look down and there's one sitting on my foot, but.... eh. After learning and understanding, I've gotten used to them, and there's nothing scary there now.
Honestly though, I've always kinda wondered at the psychology behind the common fear of insects that many people have. It's always seemed odd yet fascinating to me.