Kitten litter training

My husband brought home a kitten the other day. He has grown up with them, but doesn't appear very knowledgeable about their behavior and why they do certain things. I have always been around dogs therefore I will probably ask a lot of stupid questions because cat behavior is a foreign concept to me for the time being, so please excuse my ignorance.

1. the kitten is 3 months old an all he ever wants is to be held. Wonderful! However, he keeps peeing/pooping on me. I scoop him up, and he goes, I hold him to me, and he goes, I don't touch him until he's poop and peed in the liter box, and still end up getting at least peed on! It's getting frustrating! (He's going to the vet in a couplw days, so he will be checked for a UTI and basic health) How do I teach him to at least ask to get on the floor before going, or ideally, go to his litter box 6 feet away? (I've worked with puppies, baby ferrets, baby rodents, bunnies, all of them have asked to be set down before going, so this just going on me constantly doesn't seem right to me). The silly thing is, he uses the litter box if he's on the floor though, so he knows what the liter box is for and where it is.

2. Why does he not clean himself up after going potty? I'm getting very tired of wiping his butt every hour. Will he learn to do this as he gets older? Because he cries when it's dirty until my husband or I clean it up (which in turn upsets my Sheltie because the baby is crying/upset, so he cries and whines until it's fixed haha). 

Thank you in advance for any help/advice.

He does sound like he may have an infection of some sort.  He may be marking you but the symptoms don't really add up for that.  See what the vet says before deciding.

In the meantime, try not to startle him.  When you pick him up, do it gently and slowly.

At three months, he is just barely old enough to be away from mom.  However, it does sound like he might not have been taught properly.  Was he taken from mom too soon?  Two of our cats lost their mom at 3 weeks and we bottle fed them.  Not knowing what I was doing, I didn't teach them how to clean themselves, so they often have to have their faces washed now.  And they do not like it but I can't stand to see them with "eye boogies."

Wipe his butt with a warm, wet washcloth.  It will simulate his tongue and may help him learn how to do it himself.  Cats are really clean animals, so again, he may have something health-wise going on.  Ask the vet about that, too.

Welcome to TSC!

He was found as a stray and at the Humane Society for a few weeks before we got him, so it's very possible that he never learned. I will try wiping him down and combing him out regularly. Any tips for making it a good experience? He likes it at first, then gets bored and protests.