Hi,

I don't know whether or not nickel electrodes are "universal", but as far as covered electrodes for the SMAW process, the electrodes should have compositions similar to the base metals for which they are intended. They normally have additions of deoxidizing elements, such as titanium, manganese, and columbium. These elements also help prevent weld metal cracking. Nickel alloy electrodes are divided into five families, namely Ni, Ni-Cu, Ni-Cr-Fe, Ni-Mo, and Ni-Cr-Mo alloys. Each family contains one or more electrode classifications based upon the chemical composition of the undiluted weld metal.
Nickel covered electrodes are used to join commercially pure nickel in wrought iron and cast forms to themselves and to steel. Nickel-copper electrodes are used for welding this family of alloys to themselves and to steel, as well as dissimilar metal joints involving carbon steel, stainless steel, nickel, and nickel based alloys. Nickel-chromium-molybdenum electrodes are used for welding alloys of similar composition to themselves and to steel. Nickel-molybdenum electrodes are designed for welding nickel-molybdenum alloys to themselves and to other nickel, cobalt, and iron base metals. Limitations such as position and base metal thickness will vary with the different types of electrodes. Also, several electrode classifications are used for the surfacing of steel components for specific corrosion resistant applications. Nickel alloy bare rods and electrodes are divided into the same families as covered electrodes, but there are more classifications in most families of bare filler metals.
To address your sensitization question, there's a link to a good article at the end of this post, which seems to indicate that the problem is in the HAZ, and not the weld metal. In part, it states that the corrosion resistant characteristics of stainless steels may be adversely affected by the sensitization process occurring in a certain temperature interval from about 600 to 900 C (from about 1100 to 1650 F) which promotes the precipitation of chromium carbides at grain boundaries and the parallel loss of anticorrosive chromium from the base metal. This range of temperatures occurs naturally not at the welding stainless location, where temperature is higher and lasts only for a short time, but in two strips of metal on both sides of the weld bead. This is the so called Heat Affected Zone (HAZ) where the harmful effects take place. In a sensitized joint the chromium, which is the main "stainless" ingredient, becomes sequestered or taken out of play and locally unavailable for the protective action. If not addressed correctly, welding stainless 18/8 steels may thus cause the loss of their protective property along sensitized paths. The welded material becomes prone to intergranular attack in a corrosive environment. The article goes on to state that there are three strategies which are usually employed to oppose this tendency. In short, one is to use a very low carbon version (i.e. 304L where L stands for low-carbon) where not much carbon is available for making chromium carbides, another is to use a different type of base metal including an amount of titanium (Type 321) or columbium (also known as niobium) (Type 347) which tend to form readily titanium carbides (or columbium carbides) (and by this action the carbon becomes unavailable for chromium) at sensitization temperatures leaving the chromium free to perform its anticorrosive task, and the third strategy is to perform a solution heat treatment at elevated temperature (about 1050 C or 1900 F), for repairing a condition of corrosion susceptibility.

http://www.welding-advisers.com/Welding-stainless.html