Rosie I have the same level of sensitivity to products that make me break out in acne too, it's frustrating! [Mineral oil is THE DEVIL, lol!] I am allergic to dial soap though for some reason, so I use the softsoap antibacterial (it's kinda orange colored) instead. I also found that for non-antibacterial soap I can use liquid lever 2000 and it doesn't make my skin break out and isn't too drying either.
As far as the acne topic, I used to always get that before I found bath & shower products that don't make my skin break out everywhere. Now I just get breakouts in areas that have pressure on them (buttocks, top of thighs - from crossing my legs) since the weather is warm, and on my upper back for apparently no reason. I found that they make these benzoyl peroxide wipes, I think they're from stridex? and those help a lot. The salicylic acid wipes seem to be too harsh for those parts of the body and cause redness and itching. Scrubbing with a loofah helps too since it keeps dead skin from clogging pores and causing acne.
For moisturizer, aloe gel or aveeno lotion don't seem to cause more breakouts.
Hope that helps, wednesday!