The exception that proves the rule.
What rule? There's no specific way in which "frequency" of phasers, torpedoes or shields plays a role in Trek drama, save for making the weapons have increased or decreased odds of penetration. And that tells us less than nothing about whether the usual beam weapons are laserlike, particle beam -like or perhaps based on the transporter principle.
This directly implies EM radiation, since "bandwidth" is a term that applies to little else in this context.
Why? "Bandwidth" is valid terminology whenever "frequency" is - it is a range of allowed frequencies. There's nothing related to EM radiation there, as for example sound has frequency and bandwidth. A burst from a machine gun has frequency and bandwidth, for that matter.
Create radiation? How could we tell? There's a big flash of light, there's death and destruction. How can we claim there is no radiation?
I actually wouldn't be surprised of every vaporized humanoid leaves a small pile of ash (off camera) on the exact spot where he was standing.
That, or then ash blown to the air in a fine aerosol thanks to the heat wave. No prob with that. But the act of change from solid/liquid body to no solid/liquid body must be one of phase change. If not from solid/liquid to gas/plasma, then perhaps to some more exotic phase. In the former case, the magnitude of the phase change energies necessarily means a heat wave, which we don't get. In the latter, the side effects, waste energies and whatnot could go to the same phase where the victim went, providing us with the neat and tidy (hand) phaser effect we see.
What the hell? How can you claim phasers don't do FTL, when they demonstrably hit targets at warp? And "overlapping warp fields" can't be the solution, because we learn from other sources how difficult and even dangerous it is to overlap the propulsive warp fields of two starships.
Some sort of a dedicated FTL field unrelated to the clumsy propulsive ones can of course sheath the phaser beam, in which case it could go FTL regardless of whether it consisted of laser radiation, electrons, or perhaps shiny ball bearings. But at that point, the fact comes to play that the exact effect we see moving at FTL in starship combat is seen moving at paintball velocities in ground combat. Why complicate this very clear-cut idea of phaser beams as STL glowing effects that can be shoved to FTL by scifi trickery, with the arbitrary idea that they are in fact lightspeed beams that are
both slowed down to paintball speeds by scifi trickery,
and accelerated to FTL? Getting things to move STL doesn't require trickery unless one desperately wants to use it, so why dig the big hole of claiming that the phaser is a laser and
then try to climb out of it by saying that some aspects (really, all the demonstrable aspects!) of that laser manifest as STL visuals?
STL:
Only twice, already covered.
Nope, the effect moves STL
every time. We can
see it do so. The two episodes where time was being tampered with only serve to make this fact more blatant.
High FTL:
Not demonstrably in the entire history of Trek.
Except for all the warp battles between starships. TNG was a curious exception to this, apparently because it was too expensive to do both the TNG warp effect and the TNG phaser effect in the same shot. All other shows have demonstrated warp-to-warp firings of phaser beams, and some have shown warp-to-sublight and sublight-to-warp firings.
A side note:
Some days Voyager can scan planets for life forms fifty light years away, other days they have to be in orbit of the thing just to tell if it's Class-M.
We should probably consider that long range scans may have high resolution if they aren't realtime. That is, our heroes could check on lifeforms by passively watching a planet via a good telescope; if it had them 17 years ago, it's worth assuming there are some there now as well. Realtime measurements might require the fancy "FTL radar" beams that may lack the resolution, though.
Because the actual speed of phaser beams is never STATED to be anything other than C.
But it is never stated to be c, either. So why not pick, say, 500 m/s, when it better reflects the visual reality?
The Tech Manual describes phaser range as approximately one light second and gives every indication phaser beams propagate at the speed of light.
That's starship phasers, not the phaser as a phenomenon. And hand phasers tell us additional facts about the phenomenon, such as that it's speed is not c in the usual case.
...would be better off replaced by an atmosphere-specific type of weapon (I always assumed this was why Klingons in the TOS era used sonic disruptors instead of phaser-like weapons as their Starfleet counterparts.)
Lasers don't do well in atmospheres, either. And they are more suspect to agents introduced in the air than particle beams are, that is, to what might be "active defenses". Particle beams suffer pretty much equally in clean and dirty air.
As for Klingons using sonic disruptors... hardly. Just because their sidearms look a bit like Eminian sonic guns is hardly proof of anything, especially when Klingon weapons demonstrate phaserlike, un-Eminianlike behavior in every turn. (John M. Ford's take on the Eminian issue in
Final Reflection is delicious: according to him, Klingon weapons exports benefit from the fact that translators usually blur the lines of different sorts of disruptors, making cheap and weak sonic ones sound more attractive than they are.

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Timo Saloniemi