Now I've just received a couple of handles from a well known handle company. Unfortunately they were sent lacquered though I requested that they not be. Also one had clear heartwood and one had a wedge that is too small for the eye
Upfront I'll say you're gonna get MY opinion. I may just do some things differently, that's all. I'm still learning, too.
In the industrial process you may not be able to get unlacquered handles from some companies. It would involve taking a handle out of the flow of work -- which may or may not be real easy for them. Just sayin'. You can take the lacquer off with a spokeshave a lot faster than with sandpaper. It's my favourite tool.
Heartwood and weird grain orientations go with the territory, unfortunately. Which is not to say you have to buy them, if you have the choice and can high-grade a display of axe handles in a brick and mortar store. Buying them online will usually mean you get the first one on the hook in the warehouse. But for $10-$20 bucks a handle you're not going to get too much attention to detail from somebody. If you find a freshly-filled display of handles, just buy all the ones that look good. You'll be surprised at how few handles that really is, however.
If a wedge isn't right I make one. I usually grab a piece of maple out of the firewood pile, split off a piece and work it down with a hatchet and shave.
The first thing that strikes me about these is how thick they are! They are like clubs.
That they are. There are various reasons for that. Generally I tend to look at any handle on the rack as "a good start."
My attempt at a handle was done with white ash, because that was what was available. Since I've gotten the two I ordered (plus the one I picked up from a country store), I've learned that hickory is much more difficult to work with: it does something called "tearout" when you try to whittle it.
Ash is a good handle material. I looked at getting a chunk of quarter-sawn ash with good grain locally, enough for four handles. The cost of the wood was equivalent to about $30 per handle (counting waste and off-cuts). I'll go that way eventually, but for now I can get a $15 hickory handle and do a little work and it's fine.
Depending on the handle shape and any curvature, you will have grain ending at weird places. The original wood grain probably wasn't all that straight to begin with. Yes hickory tears out. You can minimize that by changing direction with your cutting tool. Bad grain one way is good grain 180 degrees different. Kind of like people.

What tools do you use to prepare the neck (not sure this the right term for the part that fits into the eye)?
Spokeshave and wood rasp.
Should the shoulder gradually taper into the neck or should their be a distinct edge/ledge?
"Fast taper" for me. I'll drive the head right down onto it and I want that sucker to stay there.
how slender is too slender, how chunky is too chunky?
What size are your hands? Do you have any disability like a missing finger, or nerve or muscle damage in one hand? If it feels right, go with it. If it's not right and you can fix it by taking wood off, do that. If it's too thin you'll be getting another handle. A good argument for going slow until you figure out what you like. Handle lots of old and new axes and see what's good for you. Also, you'll change your idea of what's "right" over time as you do more with it. That will affect the handles you're doing five years from now.
how wide should the kerf be?
Wide enough that you can hammer in plenty of the wedge, remembering: You can thin out a thick wedge if the kerf is a little narrow, and you don't want the wedge to bottom out in the kerf. Wedge and kerf are two separate variables, and I usually want 2/3 to 3/4 of the kerf filled with the wedge.
Where do you get your handles?
Local stores. I buy what's available of good quality. In my area that's Link-Seymour imported from the US; everything else is un-named overseas production.
How do you modify them (what do you do to them, and what do you do it with)?
Here's a link to a post on my blog (Old Hippie's Garage) about spokeshaves. In it you'll find a brief discussion about handles, too. I'm a little different in how I treat a DB handle. I hate swells and prefer knobs. I like thin handles with pretty much stock depth. I usually shave down a DB handle to between 3/4" and 7/
How close does the eye size of the handle have to be to the eye?
I prefer it to be larger, so I can work it down to fit.
(above: is this gap too big?)
I'm gonna say "Yes." At the front, anyway. I'd like to see the end of the handle in the same general shape as the eye in the head. As noted earlier, for an eye that shape you may need a maul handle.
Are there ways around tearout with hickory?
Better tools.

Don't be afraid to work slow, and try the fit to your hands a lot. Pay attention to how the handle feels when you move with it. Accuracy is more important than power of stroke, so take it slow and see how it feels. This is your personal custom grip, just like a custom pistol grip. If you see somebody doing something new and different to you, ask why and then try it out for yourself. The way I do SB handles I got from a custom handle on a hand-forged axe a friend of mine owns.