Generalities
First off, there are several generalities stated here that I do not agree with.
One of them is which action type is the most accurate. Each rifle and ammo combination is an individual, and sometimes, will perform beyond, or below the accepted "general" run. That being said, here's the way I see it...
Bolt action rifles, as a group, are the most accurate. This is because of the smaller number of moving parts, the simple trigger system, and the overall rigidity of the rifle.
Single shots are often quite accurate, but as a group, have a lower accuracy potential because of the two piece stocks, and some other factors. Again,individual rifles can be quite accurate, but as a group, not as accurate as bolt guns. Note that benchrest shooters, where accuracy is the prime consideration use bolt guns. Usually single shot bolt gun.
The bolt gun has the ability to chamber anything, and everything. Pumps, levers, and semis are much more resticted, with only a few models being able to take rounds like the .30-06. Bolt guns also have the advantage of very powerful primary extraction. If you are going to go the the max with your loadings, the bolt gun is the way to go.
I won't question how fast anyone is with their preferred action type, but my experience is that generally, for speed of aimed shots, it is semi auto, pump, lever, and then bolt action. For 5 shots, with someone experienced with each type, its only about 2 seconds or so difference between fastest and slowest. The primary reason being that for lever and bolt, one has to remove one hand from the shooting grip to work the action. With a pump, one does not.
The two main types of pump found today are the Remington (still made) and the Savage (long out of production). There are also old Remingtons and some other pump rifles on the used market, but if you are considering new, Remington is the choice.
Some pumps will shoot quite well. Most will shoot well enough. A few won't. This is an individual thing, just like any other rifle type. In general, pumps do fine for big game hunting, but are not usually good enough for varmint hunting.
An exception to this is the .22 cal, where the pump often does just as well as anything, and is faster than everything save the semi auto.