We know very little about the limitations of the transporter despite even getting snippets of dialogue specifically referring to those. What we do learn is interesting bt possibly misleading in all its ambiguity:
1) Beaming down typically involves stepping exactly onto one of those pads and waiting for an operator to operate the controls. In the second pilot, an unconscious transportee is even walked to a pad and pushed to precariously teeter on his unconscious legs, without support, so that his beam-down can commence! Yet sometimes people can beam down while not confined to a single pad - say, two people can hold hands ("Journey to Babel"). So the "limitation" about using just the pad seems arbitrary.
2) Beaming up involves very little beyond the operator operating. It's impossible if the ship's shields are raised, though ("Arena"). And to get the action started, one typically needs to signal with a communicator, although use of sensors may do the trick, too ("The Enterprise Incident").
3) Beaming down through somebody else's shields is also impossible ("Dagger of the Mind"). Yet beaming down through the ship's own shields seems simple enough! Ambassador Fox and his aide do that seemingly without assistance ("A Taste of Armageddon").
4) Beaming from A to B without using the pad in between seems impossible ("The Cloud Minders").
5) It's risky to beam from within the ship to within the ship for some reason - proximity, confined spaces, other? ("Day of the Dove")
6) Generally, only up to six people beam down - if there are more, they arrive in several batches of no more than six ("The Apple"). But more can beam up at once ("Day of the Dove"). So arbitrariness again.
7) Beam-down can be done without an operator, by setting the controls on some sort of a timer. If there's nobody at the controls, though, beam-up appears impossible, there being no remote control option ("This Side of Paradise").
8) There's a decontamination option available ("The Naked Time"), but all sorts of stowaways still find ways to travel in the beam ("Man Trap"), suggesting there isn't much automation involved in decon.
That's already plenty of rules or seeming rules, although it's nothing compared with the complications introduced in the spinoff shows...
Timo Saloniemi