I will toss in my 2 pennies on the subject of live centers.
I use a stub of soft steel in the 3jaw as a live center. I cut it fresh each time I use it, even if I had a tool post grinder, I doubt I would grind it.
This works well for me since I use the 4jaw almost exclusively when I put work in a chuck. My chucks are well fitted to my spindle (D1-4 nose), so swapping them back and forth does not seem to cause any issues, though I will often recut the center just because it is "best practice."
I have had some parts go on and off the lathe upwards a dozen times as they either get pushed aside for another project, fitting up, or for milling operations, and they always go back on as concentric as I can measure with my DTI.
Like mentioned above, I like to do my roughing work with the material in the chuck, and try to limit my DOC between centers to under 0.070"
As for the other stuff. A sleeved #3 is just as accurate as a #5 center assuming it has been clocked. A #5 rolling center is an order of magnitude stronger than a #3 center, but is also substantially fatter and that much harder to get a tool around, and you can give up all that stiffness using an extended center. On the other hand repeatedly swapping the sleeve around tooling does cause wear and tear on the sleeve.
I also fall in the "old school" camp that defines a dead center as the thing that goes in the tail stock, a live center is the smooth pointy that in a headstock, and a drive center as a pointed thing with spurs that goes in the headstock (seen mostly on wood lathes). If it goes in the tail stock and rotates passively with the work, it is a rolling dead center. I don't get hung up about it though, and often refer the rolling dead center as a live center, so long as folks know what I am talking about there is no problem