For example if a full health Multiscale Dragonite switches in on Stealth Rock, will it do the normal 1/4 damage or 1/8 because of Multiscale halving the damage of a super effective attack?
Nope it takes the regular 1/4 damage, as multiscale doesn't protect the holder from passive damage, such as status and entry hazards, but only halves the damage from damaging attacks.
Source: experience
Also, check this out: http://bulbapedia.bulbagarden.net/wiki/Multiscale_(Ability)