eggshell I find holds the dirt and will never look fully clean, especially where it is handled, such as doors and handrails.
satinwood is easier to clean, gloss easiest.
Oil paint is more durable, especially outdoors, but water paint will do indoors. it is I find less glossy than oil paint.
White oil paint tends to yellow if it can't see sunlight.
For prep, clean well. If surface is perfect and you are not changing colour, one topcoat will do. More often, fill any chips, lightly sand with fine paper, hoover off the dust, apply white acrylic primer undercoat, examine carefully, wipe lightly with fine paper to remove grit, spiders legs or runs, touch up and smooth off u/c if necessary. Apply another u/c if the old colour shows through. For skirtings, pull back the carpet and hoover out the gap well, and use a piece of card to prevent the brush touching the floor or picking up fluff. Wipe off all marks or drips before they start to dry.