1) Only if Short Sword is a Monk weapon. You could however TWF a Kama with Unarmed Strikes from a Flurry in one Full Round. The Kama would be all your offhand TWF strikes and the Unarmed Strikes would cover your primary weapon in the TWF sequence.
I specifically chose a non-monkish weapon because if you were using, say, a kama, it would be considered part of the flurry.
2) Yes they require proficiency because they don't explicitly say they don't require proficiency.
That's circular logic. "It doesn't say we
can't do it, therefore we
can." Most DMs hand-wave proficiency with the club, even though it's listed as a simple weapon - it's not that hard to swing a large piece of wood.
3) No, they both can be done "as part of a full round", but do not consume the full round.
PHB, page 143: Full Round Actions. "A full round action requires an entire round to complete. Thus, it can't be coupled with a standard or a move action, though if it does not involve moving any distance, you can take a 5-foot step." Full attack is listed there. Therefore, it takes up the entire round.
It doesn't, you take -4 due to non proficiency if you don't have proficiency with Unarmed Strikes (and Monk's get it by default by having Improved Unarmed Strike feat even tho it isn't explicitly called out in the monk description.)
Um... ALL characters are proficient with unarmed strike - it's a natural weapon, and all creatures are automatically proficient with their natural weapons. A monk (or anyone else who takes the IUS feat), however, is the only one who can deal lethal damage and thus make armed attacks instead of unarmed ones.
Well, I guess it depends on which "why" you're asking.
Why do gauntlets require proficiency from a rules perspective? Because gauntlets are listed in the
SRD - Equipment Weapons section of the rules. ALL weapons in this section fall under the category of simple, martial, or exotic, and gauntlets are specifically called out in a table as being simple weapons. Further noted in this section, all characters must have the relevant proficiency, or take a proficiency penalty. So basically, gauntlets require proficiency because the rules say they do.
Maybe they're listed there because they don't fit anywhere else. Unarmed strike is also listed there, but all characters are automatically proficient in it.
Why do the rules call out gauntlets as weapons? That's an interesting design question that I don't have a great answer for. Note that not all things that cause damage have to be called weapons. Items like caltrops, alchemists fire, and thunderstones all fall into a category that would be called "weapons" in normal english usage, but are not classified as weapons in D+D terminology. All of these items can be used by anyone without any proficiency issues.
That's because they're area of effect weapons, like grenades - you only need to get a grenade
close to the target for it to be effective.
No, they are not. TWF and flurry require that you use the Full Attack action, which is a full-round action; they are not special actions of their own (like, for example, Charge). The distinction is important, because if flurry were its own special full-round action, you couldn't combine it with any other full-round action (such as a Full Attack using TWF). But since TWF and flurry are each performed as part of a Full Attack action, both can be performed during a Full Attack action.
I agree with the first part, but I'm not sold on the second. If they're both full-round actions, it doesn't make much sense to me that they could be combined.
Not quite true. You can't make any attacks with the short sword while flurrying, but nothing prevents you from wielding it.
That's putting a pretty fine point on it.