Along with PinkPiranha's suggestion -- the
DHC line of skincare products is based on pure olive oil, so you don't have to mess with the thick made for cooking kind. People swear by it, especially sensitive, dry, and mature skin types. I have used it myself and it's never caused an allergic reaction on me but I have the opposite skin type --very oily so it doesn't do much for me.
For badly chapped or irritated hands, there is a German hand cream that works great. It's in a green tube and they sell it at Sally's beauty stores...I can't remember the name right now. It may smell a little funny but it absorbs very fast and acts as a bit of a barrier as well.
As for the sensitive eyes, I have sensitive eye*balls* I know that sounds gross, but I had laser vision correction so I have chronically very dry eyes (sometimes painfully so) and I'm a bit paranoid about eye make-up. All that tugging with eyeliner and removal
really take a toll (not to mention accelerate wrinkles). I didn't start wearing eye make up as much until after the lasers.
Now I try to use Almay eye makeup, as that seems to be the only company who really goes out of their way to make 'safe for sensitive eyes/contact lense wearers' grade make-up. For removal, I try not to tug and instead use a cream or oil based cleanser and just glop it on my closed eyes and tap it in and wash off with lukewarm.
Eyeliner and mascara are especially tricky, and will not totally come off with just washing off. I use a cotton swab with petroleum jelly or other 'eye safe' gel/cream/oil and dab it on, let it sit, then dab it off.
But honestly, if I see any redness in my eyelids after make-up removal I give it a rest for at least a few days.
I've heard that professional actors develop a lot of skin problems over the years due to constant, prolonged, and heavy use of makeup (especially cream/grease stage makeup)
excema is an horrific condition, as is asthma, and any allergies. 
