I want to convert from my quill stem to a modern threadless. Would all I need be the following? Obviously, I'm excluding the stem itself.
Brand-X Quill Adaptor Alloy | Chain Reaction Cycles
M Part Splined Headset Spacer | Chain Reaction Cycles
would the spacers sit flush? Or would the stem be wider than the 1" spacers and would I need to buy a new top cap?

And should I buy a threadless headset?

Keep your threaded quill stem. For ease of height adjustment, it can't be beat. If you need a taller stem, go with a Nitto Technomic.

About the only thing a threadless stem has going for it is you can more easily swap out bars but getting a height extension with them sucks.

Sometimes old really is better than new.

I just bought a threadless adapter so I so make use of the stem, handlebars, Shimano Sora brifters, and Tekro interrupter levers that were on the Felt F90 that I parted out.

I'm building up a Trek 1400 frame that I got off eBay, and although I already have a quill handlebar stem and handlebars for it, I wanted to try the Sora brifters, and the Tekro interrupter levers without removing them from the Felt handbars (different diameter from the SR handlebars and stem that I already had on the Trek.

By buying a chromed quill threadless adapter from the local Performance Bicycle Store for $14.99 (same price as online from China), I was able to drop the complete Felt F90 handlebar assembly onto the Trek, without having to wait for the adapter to be delivered from China.

Now let's look at your situation, and your questions.

The Performance threadless adapter comes with the adapter body, a quill wedge, a top cap, and the long bolt for the quill locking wedge. It works the same as a quill stem locking to the steering tube to hold it in place.

You do not need to buy a threadless headset, as you will still be using your conventional threaded headset. You do not need any spacers, because the threaded headset is still going to be holding the fork in place. Spacers are necessary on true threadless system to take up the extra fork steerer length so the headset can be secured. Since you are still using a quill system, the spacers are not necessary.

The threadless adapter only has around 1" of vertical adjustment. If you need more rise than that, you need to look at threadless handlebar stems with a different angle or rise, OR buy a threadless extension adapter which will attach to the top of the threadless adapter. Adding height to the handbars is a PITA with threadless headset systems. It's not like the ease of buying an extra long/tall quill stem.

The cap that comes with the threadless adapter is a smaller diameter than is used on true threadless stems. The cap is trapped because the quill bolt can't be removed. Removing the quill bolt and the cap would drop the lock wedge. The threadless adapter cap is not used to set the preload/tension on the fork and headset. The larger diameter of the cap on a true threadless system is to keep the handlebar stem from sliding over the cap before the stem bolts are tightened. The adapter cap is a smaller diameter so you can slide the threadless handlebar stem onto the threadless adapter without removing the threadless adapter and sliding the handlebar stem on from the bottom of the threadless adapter.

So to go to a threadless handlebar stem on your threaded headset bicycle. All you need is a threadless adapter (as long as the limited rise of about 1" is okay), a threadless handlebar stem with a clamp diameter that is compatible with your existing handlebars, and some handlebar wrap to rewrap your handlebars if you need to unwrap them to move the handlebars to the new threadless handlebar stem.

If you do you'll need to buy a new fork too

That opens a whole new can of worms.

The vintage roadbikes all had 1" forks. Vintage MTBs had 1 1/8" forks.

You can, of course, get 1" threadless forks and headsets. And maybe 1" spacers. But, then the selection of 1" threadless stems is limited, so you'll probably end up with a 1 1/8" threadless stem and shims.

Anyway, if I was going to replace the fork anyway (steel ==> CF, building up forkless frame, etc), then I might consider converting from original threaded to threadless.

Otherwise, I'd probably stick with the threaded + adapters.

Or, just use the quills that have worked for ages. I put a winged handlebar on one bike necessitating a 2-bolt open faced stem, but somehow I accidentally found a 26.0 winged handlebar (which apparently is relatively rare now). And, there are a few quills that take the 26.0 bars with a 2 bolt face.

Trek supplied 1" threadless forks on the 1993 2300 Composite bike. The only model year (and model) they did a threadless fork on Composite bikes. They also supplied some threadless 1" forks on some model year carbon bikes (but never on the bonded aluminum bikes from what I've found).

I played with the idea of converting some of my Trek bonded aluminum bikes to 1" threadless using original Trek threaded forks from larger frame bonded aluminum bikes. I figured that if I could find the fork from a really big Trek bonded aluminum frame, the steerer tube would be longer enough for me to hack off the threaded end of the steerer tube, and still have acceptable length on it to attach the threadless headset and stem with spacers (this would yield the largest choice of paint colors, plus the choice of chromoly or aluminum fork blades). Or maybe get the threadless 1" fork from a 1993 Trek 2300 (pretty rare and only came in one color) that is the same size frame as the Trek bikes I would be looking to convert.

Profile still makes 1" quill stems with 26mm handlebar clamps and 2-bolt clamps. I have one (in black) that the PO of my 97 Trek 1400 installed. From what I've seen they are available in polished silver (didn't see any of the black ones offered) on eBay. Personally, I think they look kinda clumsy (skinny post with large horizontal tubing and clamp), but that's my personal opinion.

I prefer the classic of the Cinelli, 3T, or Sakae single bolt quill stems.

^This.

OP, why do you want to convert? Unless there's some strange situation, there's no advantage to going threadless- Quill stems rock! (Actually, they stay nice and tight...no rocking at all ! )

All you should need is the adapter and stem. No spacers should be needed. I love the look of a nice quill stem but I just made the change recently on an older bike just to be able to use a modern bend handlebar with a 31.8 diameter.

Quill adapters can be raised/lowered just like a quill stem. Same concept.

Another advantage of a quill adapter is it's much easier to swap out stems for a different length or rise.

Just the adapter, skip the spacer.
I have done this on 2 of my vintage bikes.

Do threadless stems rock? I want more options for adjustment, because I'm growing and for whatever reason my bike isn't...