A: It's manipulation of the $ONMISSION variable (abbreviated to simply "om"). This variable dictates whether you are in freeroam mode or not, and therefore whether you can start missions or not.
For example, you can start a rampage + Vigilante at the same time ("rampage-starting") by using a saved replay, as rampages can be activated when picked up in a replay. $ONMISSION is set to 1 twice. But if you then cancel Vigilante, the game will set $ONMISSION to 0, as though you are in freeroam mode, when in fact the rampage is still running.
Now you're able to start any mission while the rampage is still active, and then fail or complete the rampage, at which point the game sets $ONMISSION to 0, meaning you are not considered to be on a mission. This basic mechanic is heavily abused to induce instant mission passes (or instapasses), as well as other effects that result in quicker completion.
A: Full-game speedruns of Vice City are typically done on the Japanese version because it is one of the so-called "Haitian-friendly" versions (look it up if you don't know what that means). This means the mission script has been slightly modified from the original game, which in turn is relevant for different instapasses (see below if you don't know what that is).
Why specifically the Japanese version, you may ask? Well, the other Haitian-friendly PC versions have some issues. They are:You can use the Australian version for No SSU since Ammu-Nation isn't used in the route, but there is no practical advantage in doing so.
A: It's shorthand for instant pass, which is further abbreviated to "IP". It refers to instantly passing a mission by starting it while on a certain part of another mission. This requires you to be om0, an unintended behavior.
For example, if, on a Haitian-friendly version of the game, you start Distribution while on the first part of Hit the Courier, Distribution will pass.
It just so happens that the part of Hit the Courier's mission script before you obtain the plates has about the same offset into the mission code as the mission pass section for Distribution on Haitian-friendly versions, and for some reason starting another mission carries over the first mission's script offset. Thereby which it is instapassed. (Here's a visual example courtesy of Patrick)
There are quite a few different instapasses, many of which can be observed in All Missions and 100% runs.
A non-exhaustive list of the instapasses you may see (many are 'Japanese-exclusive'):
A: You can pick up objects within replays.
A: More replay stuff. When you change movement animations and play a replay at the same time, Tommy will begin sliding, which works during phone calls and is very useful for heavy weapon rampages, since you go a lot faster than the forced slow walking speed with those weapons.