1) This totally depends on your play style. The Cleaver is pretty middle ground, with the Axe doing more damage with longer attacks and the Cane being quick but less powerful. There's absolutely nothing wrong with sticking with a starting weapon all the way to the end because, with upgrades, they all end up effective. Also, after the second boss I believe, you unlock the ability to easily buy any of the other starting weapons of you want to give them a try.

2) Once you replace your starting armor set, the rest is mostly cosmetic. The one the vendor has is the same as a set you can find in the sewers. It shouldn't be too hard to look up its location online. Once you've got that set, you rarely need to replace it unless you want more immunities or to look cooler. I also don't believe that you can upgrade specific armor pieces. The upgrades are either applied to individual weapons or your character as a whole.

3) Dodge frequently. Spend time with a single enemy just to get a feel for dodging his attacks. Look for shortcuts. Checkpoints are sparse but shortcuts effectively accomplish the same thing by opening gates or turning levers. Shooting an enemy in the middle of their attack animation allows you to to run up and counter them with R1 after hearing a certain sound. After dying to the first boss you unlock Kyle ability to level up. Do this frequently. Always stock up before re-attempting boss fights. And finally, to help you on the first boss, take a moment to look at the shape of its body. Its design indicates a favorable side to dodge on.

Good luck, and enjoy! It sounds weird to say, but patience and practice pay off a lot! Don't get discouraged!