The culprit in most soaps (barring a reaction to something else in the soap) is generally the fragrance (both essential oils and fragrance oils can cause reactions - but its more common with FOs - however QED soaps have very pronounced scents and contain a lot of EOs which have caused some people to experience irritation).
Unscented is best, but chamomile and lavender EO scented soaps work well as both of these plants have properties that sooth the skin and are often included in skincare products. I'm not sure what Honeybee uses, but I do think Colleen makes a nice lavender soap (FYI, lavender does not smell sweet or flowery - but actually has a very crisp, green, herbacious scent that is more reminiscent of rosemary than roses - a very masculine scent).
In generall, the less heavily scented the soap, the less likely it is to cause irritation.
Also, for me, the combination of glycerine, shea butter and clay just can't be beat - the glycerine and shea butter provide great moistuirizing properties and the clay adds slip that helps the blade glide over the face. I've only been using Honeybee soaps for a couple of weeks, but they're betther than anything else I've ever used (I've not used Mama Bear or Colleen's soaps, both of which are supposed to be great as well and are on my list).
I like the alcohol-based splashes - they actually help prevent infection and I like the burn - I don't think they contribute to the type of reaction you experience as the burn stops as soon as the alcohol evaporates.
You might also want to experiment with different types of post-shave treatments. Witch hazel is nice and soothing, also, Jason makes some nice gels (witch vera and tea tree) that contain aloe, witch hazel and other good stuff - very light, soothing and inexpensive.
I strongly doubt that your reaction has anything to do with your blades not having been disinfected - unlike DE razors, there's really nowhere but the pin area for stuff to accumulate and that doesnt' come into conatact with your face. If you do want to clean your blades, I suggest a quick wipe with rubbing alcohol or a 10% bleach solution - avoiding the scales (but you might want to get some in the pin area if you're particularly anal. I've never done this with any of my razors and have never had a problem (however, I would always scrub and disinfect used DE razors given the numerous places where crap and crud can accumulate/grow).
As for the oil you use on your blades being the culprit - a lot depends on what you're using. If you're using a light coat of mineral or camilla oil, I strongly doubt that this could be an issue - however, if you're using something petroleum based or synthetic, this could be a possibility. In any case use a very light coat and take a soft damp cloth and wipe it off before stropping - alternately, a quick rinse in hot water should do the trick.
I don't use oil anywhere but occasionally on the pins - unless I'm storing a razor for an extended period. If you're really worried about rust, I would make sure your razors are in a breathable container (anything but plastic, glass or metal should do) - if you're still worried, buy a silica dehydrator (the same stuff that gets packed in with cameras/optics in little pouches) and put it where you keep your razors - they sell them at Lee Valley for very little $$ and they're treated so you know when the silica is saturated - at which point you put the can in the oven to recharge the silica.
Hope this helps.