I am probably in the minority here, but a folding saw and small fixed blade knife are my recommendation over an axe. Smaller, lighter, faster. If you pick wood with purpose, you don't really need to split wood. You also don't need the classic open campfire. It looks nice, but you lose a lot of heat for the wood you go through.
I like my ENO hammock, but I will probably upgrade at some point soon. A simple tarp over top helps to cut down on the wind which will suck the heat put from under you.
Use a poncho liner as an underquilt, wool blanket for bedding. I'd call it a 3 season setup. Maybe 4 season here if you pick non extreme nights.
Bic lighters are the best place to start for firestarters. I always have one on me and a couple more in my pack. I carry a firesteel as well.
One night camps allow you to get away with not much gear. You can go lightweight or use the extra space for a nice supper and breakfast.
As far as a bug net for the hammock, I just picked up a cheap one from the surplus store that I adapted to work. You already have a ridgline run for your tarp, so it really isn't difficult to set a net up.
Obviously bring some sort of small first aid kit. And most importantly, have fun.