I decided to break this into two posts to keep it easier to follow. As to the number of heats resulting in distortion- only if it produces strain effects and, more importantly, uneven strain effects. Just heating the steel could induce, or reduce strain energy, depending on how you cool it. When you bring a steel to glowing you have thrown more energy at it than all the bending and hammering you could ever do, but the conductive nature of the metal makes it want to distribute that energy evenly, so we have to do something wrong in our heating to overcome its self regulating behavior. Heating very unevenly will do this, as will cooling very unevenly.
I teach a method of heating for normalizing, and other treatments, in a forge that better deals with a critical point in the heating process- decalescence. At 1335°F the steels expansion and taking on of thermal energy drastically changes as it makes a profound crystalline shift to put things into solution. The steel stops expanding and begins contracting, and the glow dims as it starts sucking up much more energy to accomplish this shift. If you have a very thin section attached to a very thick section, this lag will be exacerbated by the thermal mass differences, and the process of decalescence will fight you in evenly heating. This could produce warpage, but it will most likely result in your tip or your edge overheating before you spine comes up to temp. So I teach my students how to choke back the forge and gradually approach decalescence and, when the shadows have left the steel, then hit it with the high heat. They are always amazed at how the steel will heat completely evenly at this point with no overheating of tip or edge. The steels natural conductivity can now handle the energy without decalescence messing with it. By the way- stock removers simply call this a pre-heat or a presoak, sometimes us anvil jockeys can learn a thing or two from those grinder guys.
But, by far, the greatest potential in getting uneven strain energy is in cooling. On cooling there are several temperatures where the steel can expand or the opposite effects of releasing more energy can fight us in the same way; the more evenly we can cool the steel the better. This is why normalizing MUST be done in still air even to be considered normalizing. A blade laid on its side is a guaranteed warp.
Normalizing: This is not to be confused with "stress relieving" as that is a different heat treatment entirely. Normalizing is exactly that- it normalizes the internal condition of the steel making it uniform. Uniformity in grain size, uniformity in phases and carbide condition/distribution. Normalizing may actually add a buttload of strain energy but it is designed to add it evenly and uniformly; thus no warpage. Normalizing is your number one tool to eliminate distortion later on. If your blade warps during a proper normalization, the effects were not complete- repeat the normalizing until the distortion goes away, or it will come back later.
Annealing: Its purpose is very straight forward- to make the steel soft. All steel can benefit from it, but it is not always necessary with 10XX steels. Anything with alloying should be annealed if you are going to machine it and the same applies any 10XX above .8% C. I strongly discourage the old practice of lamellar annealing (heating to nonmagnetic and slow cooling in the forge or vermiculite) as it requires that some of the benefits of normalizing and grain refining be undone and it can be outright bad for many steels. Spheroidal annealing is subcritical and thus does not remove any effects of previous beneficial treatments. Any annealing operation done properly (evenly) will reduce distortion.
Stress Relieving: Using thermal energy to annihilate strain energy begins at relatively low temperatures, like tempering temperatures, but the effects are very thorough at temperatures exceeding 1000°F. Heating the steel to 1200°F will wipe out strain effects stored in the steel while not touching the effects of normalizing, grain refinement, or annealing. What this treatment does is resets all the 'mis-alignments" in the atomic lattice of the metal, but it cannot move the phases, alloys, or carbide distributions, so if your normalizing or annealing didn't do their jobs this treatment cannot overcome that.
Hardening: Its very purpose is to load the steel with as much strain energy as possible, so it had better be done evenly. The problem is that the variables that can produce uneven stress in the quench are infinite, so it is the number one producer of warps. This is the heat treatment that repeating will unavoidably result in uneven strain energy. The best bet is to revisit one of the previous treatments for a "reset" before running the gauntlet again.
.)