Some things attributable to lazy writing or lack of guidelines can actually be made awesomely workable -- like 3D room sensors (that is, not a single location) mapping who's moving where and how fast and such. It wouldn't be that difficult to write some sort of predictive probability-calculation subroutine (with a probability threshold set by experiment) that can tell when someone's just moving around and when they're heading for the door. And, if they stop with the door open and turn back to say something to someone in the room, the probability still remains high enough that they're going to continue out the door that they're held open. But if that person keeps talking and moves back into the room, the probability drops below the threshold and the doors close again. That sort of thing.
I can even fudge the "realtime" comm system paging by saying it's something to do with the FTL processing boosters in the main computer allowing the computer to hear the page from the speaker and play it for the intended recipient before the speaker had actually finished speaking. Same applies to the lack of signal lag from ship to surface. Take that, Einstein.
For the tapping, it's said in the tech manual that the tapping of panels or commbadges is largely habit. The one time it's most egregious is in the first-season finale when Riker is paged by Picard and walks halfway across the room to tap the comm panel on the wall, just so the writer could have the obnoxious cryo-refugee see and be able to figure out how to call the bridge later. Aboard ship or for intrafacility comms, just speaking to the air is all you need. When you're outside and receiving on your commbadge, you just need to speak in acknowledgment. When you're inside and receiving an outside transmission, you need to go to the commpanel (since it's usually a video feed). The only times you need to tap your commbadge are when you're outside initiating a call, or to end it (or put on hold), rather than saying "[surname] out".
Would've been nice if they'd been consistent with that...
--Jonah