This is from "Weldability Of Steels" by RD Stout.
Sulfur
" ... The welding of steels containing sulfur contents of .10% or more for free machining characteristics is often troublesome, ..... Sulfides present as elongated stringers increase the susceptibility of base plates to lamellar tearing. (There is much more than this with refernce to the combination of sulfur with O2 and H2 and it seems to indicate that some steels with excessively low amounts of S can be more susceptable to deleayed cracking)"
Phosphorus
"... As little as .05% results in noticable increase in yield strength and a corresponding decrease in ductility. Phosphorus tends to segregate readily. It difuuses too slowly to permit homoginization in normal heat treating or welding processes, and thus may cause harmful banding, and diffusion of carbon away from high phosphorus areas. Nevertheless, good welding results have been obtained wih corrosion resistant low alloy steels containing approximately .15% phosphorus for improvement of corrosion resistance."
There should be someone that could simplify or add to this information I quoted. I thought this might get you started. If my knowledge were greater I could possibly add more but thats not the case.
Gerald
weldinginspectionsvcs.com
One way the weldability of steel is classified is through susceptablitiy to delayed cracking (cold cracking/hydrogen embrittlement/etc...) the general formula that is used that determines a steels comparitive susceptability is this: Pcm = C + (Si/30) + [(Mn + Cu + Cr)/20] + (Ni/60) + (Mo/15) + (V/10) + (5B)
Another factor in weldability is hardenability. One way to determine the hardenability of steel, called the Carbon equivalent. As you mentioned carbon is a major factor in the weldability of steel. This formula basically, determines how much carbon would be in the steel if all the elements that act to increase hardenability were converted to carbon. The formula for CE = C + (Mn/6) + [(Cr+Mo+V)/5] + [(Ni+cu)/15]
The phosphorous and sulfer have several effects on weldablility. One is that they contribute to hot cracking of the weld metal. They also detract from the toughness of the weld/HAZ/base metal. I have never run into problems with too low of a sulfer as quoted above from RD Stout. We routinley weld on base metal that has less than .001 % S, and it usually helps our CVN tests quite a bit as compared to higher S containing base metal.