If you're invisible, you can make a check with advantage to see if you're quiet. If you're concealed by an illusion of someone else, however, you should use charisma (Deception) with advantage, opposed by 'passive' insight. It has the same effect--you're undetected, you get surprise.
There's no inherent advantage conferred to stealth by invisibility. Who is harder to notice, the invisible person walking across a room in front of you (making footprints and footsteps, knocking things over, breathing, rustling fabric, etc), or the "visible" person hiding around a corner?
Total concealment is, under most circumstances, a prerequisite for hiding. How you achieve that total concealment is irrelevant, unless noted by a particular special feature or ability. The advantage of being invisible is that you can
always attempt to hide, because you are always, visually, concealed. (And you get advantage on your attack rolls and impose disadvantage on attack rolls against you, which is also pretty damn useful.)
As for the case of [MENTION=67338]GMforPowergamers[/MENTION]'s bard and surprising a warlord by appearing to be somebody that he trusts? Absolutely. The warlord is unaware of a potential for combat. Whether this obtained through "stealth" or through other, more devious means is irrelevant. Of course, the bard has to keep the illusion up until the last possible moment.
As for multiple attacks? The way most people seem to be ruling surprise here, yes. You are no longer surprised at the end of your turn on the round in which you were first surprised. So, until that point, you are surprised, which means you'd yield Assassinate to a bard-Assassin, and, of course, you can't move that round.
Of course, the bard-Assassin would have to win initiative, otherwise, the warlord is no longer surprised once his turn is over and the bard is able to act.
(As I run things, which not all agree with,
if the bard was able to strike from hiding, and in no way give up the game until the blade actually strikes or misses the warlord, that attack would be allowed to go first, but any further attacks do to multi-attack would happen in normal initiative order. But that's kind of my house rule, I guess.)