Let me take a shot at this and the others can chime in and correct me.
First, be sure you checked the concentricity of your spindle correctly. That is, run the lathe for 20 minutes or so at moderate speed to warm it up, then shut it off and shift into a gear that allows you to easily turn the spindle by hand. Clean the taper inside the spindle so it is oil and dust/dirt-free and then use a solidly mounted DTI of adequate sensitivity (0.0005 works) and lightly preload the tip with about 0.015" of pressure before zeroing the dial. I usually place the tip at 6 O'clock so I can see the dial easily. Place a mark with a Sharpie on the spindle mount at 12 O'clock and then slowly make one complete revolution, returning the mark to 12 O'clock; the dial should return to zero. If so, then slowly rotate another revolution but look for any deviation from zero and this will be the run out of your spindle. When you do this, make sure you turn the spindle in one direction only. If you are doing this any other way then your results are going to be variable. Concentricity is NOT checked with the spindle running.
To figure out how to clock the chuck, lock an accurate test pin in the chuck and lock it down with the factory-marked pinion (as mentioned above) and check the concentricity of the pin. Normally, I put the tip of the DTI at 12 O'clock, perpendicular to the test pin and very near the chuck jaws. Do the same thing - make one revolution to be sure the dial returns to zero and then another to check how much run out you have and write it down. Now remove the chuck, turn it so the next D1 pin lines up, lock the chuck onto the spindle and re-test the run out as above. One position will give the least amount of run out and that is the position you should always install your chuck in. Most spindles have a witness mark somewhere on the spindle mount; make a corresponding mark on the chuck right across from it so you can install the chuck the same way every time.
As to your questions:
1. If you don't remove the part from the chuck and your chuck is in decent shape (jaws and scroll are okay, no major issues) then the part will have the same runout as your spindle, plus a little more due to cutting loads. It should be at or very near the spindle run out unless your spindle bearings are shot. The amount of "run out" your chuck has with a test bar has little to do with your actual results. Bear in mind that if you take the part out of the chuck and then try to get it back in place then your run out will be greatly magnified.
2. Not unless your current jaws are damaged.
3. 4 jaw scroll chucks do. 4 jaw independent chucks will have as little run out as you have the patience to dial out. It is possible to dial out almost all the run out in a 4 jaw chuck but it can take a few minutes to a few hours, depending on the phase of the moon or the current wind direction. Some days, God just smiles on you ...
4. It is important to assure accuracy when it counts. The trick is to know when that is. In the case of your 3 jaw chuck, I would just accept that it cuts adequately and move on to learning to run the lathe or how to grind a good tool.
Bear in mind that there are adjustable 3 jaw chucks called Adjust-tru chucks and they can be adjusted to run very accurately, enough to satisfy most folks. I don't own one; I use ER collets when I need that kind of accuracy.
As hobby guys we tend to obsess about accuracy and will move the earth to cut something with tenths accuracy BUT there are times when you should do this and times when you don't, and most times you don't. Set a reasonable tolerance expectation and learn to hit it. Otherwise you will spend more time fiddling and less time making stuff and that's what this is all about, right? Making stuff.
Mike