I find baby oil way too greasy to use on the eyes, I can't see for ten minutes afterwards if I use that! I like the No7 Clanse & Care stuff, or if you're wanting to splash out, Lancome Bi-Facil is great.
Vaseline and baby lotion should, I've been told, never be used near they eyes as they stop the skin from breathing and also leave the skin in a less than desireable state for reapplication.
I use vaseline, just a small amount rubbed on the lashes, not all over the eyes, and if you don't use too much you will be able to see straight away afterwards - not hazy! I also find that since using vaseline my eyes are brighter in the morning, and not red - could be that it doesn't cause allergies.
I've found the best way is to smooth a little remover over the eye area & use damp cotton wool to wipe away. Dry cotton wool or tissue just drags. I use Avon, conditioning eye makeup remover.