Grapeseed oil is one of the lightest oils, very balancing, rich in antioxidants and not greasy. (Shhh...I just use a cold-pressed one for cooking
! But it's worked the best.) Olive oil really didn't work well for me but many do swear by it. Apricot oil, sweet almond oil and jojoba (actually a wax) are other good choices, but read up on aromatherapy carrier/base oils and try some out to see what suits you because each one has very different therapeutic qualities. Unrefined avocado oil is extremely rich and waxy, you may want to dilute it with some other oil. Study, experiment.
DHC, Shu, Fancl, etc. cleansing oils all have chemicals that make them emulsify when you mix with water and wash. They left a kind of slick silicon-y film that I didn't like. Some made me break out.
I'm not saying they're bad, but they're formulated to wash away, and it's just a very different concept from wiping with natural oils.
Don't be afraid of leaving makeup residue, if you wipe away gently with a moist cotton it will be fine. Those prone to breakouts have much to gain; oil removes oil, protects the skin's moisture and natural healing, and
prevents the overproduction of sebum, which is what makes it so suitable for acne-prone skin. I've had amazingly clear skin since starting this. Again, see what oils suit you. Personally, I wouldn't wash later with water and cleanser, I find it strips away the skin's own natural protection and negates the very benefits of this method.
Anyway, hope that answers some questions.