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There was an entire TNG episode (Season 6, Episode 12) whose plot centered around this:
spoiler
Moriarty was reactivated by mistake, and took the ship hostage, demanding to be able to leave the holodeck.
Geordi and Data spent half the episode experimenting with beaming (inanimate) holographic objects off the holodeck, to no avail. With that said:
spoiler
Their transporter turned out to be a holographic fake (and so was Geordi), so who knows if the results were valid.
Both. Though regular holograms would immediately dissipate on arrival, since they're separated from the projectors maintaining the holomatter.
There have been many cases which The Doctor has become solid so other solid objects can no longer pass through them. If the object we are seeing being beamed is the mobile emitter, then is it necessary for them to be on a separate pad? I imagine the person accompanying The Doctor could just hold the emitter instead.
The Doctor needs to externally reconfigure himself through the computer control panel to change his tangibility, he can't just do it on the fly.
Transporting him as if he was a human, rather than just the emitter probably helps Voyager's crew remember that, instead of treating him as a piece of equipment.
It's also unclear whether transporting just the emitter instead of the whole hologram might risk damaging his holomatrix, since you'd effectively be forcibly removing the emitter. He wasn't designed around having a mobile emitter, or with the ability to be transported.
The Doctor has controlled his intangibility with a thought (command) multiple times. He mostly does his own matrix reconfiguring since like season 2 when he gained his autonomy.
The light is immaterial (lol) to the holograms existence. They are only solid via fancy forcefield transmitted from the holo emitters. For all intents and purposes they seem to only exist as programs within whatever computer operates the holo emitters, whatever or whatever that may be. The only limit on where they can be is the speed of the computer system and links within it. They can send holo programs between the alpha and delta quadrants, but are limited by the speed and this can only send smaller programs. The mobile emitter often gets transported, but that's because they only have one. If the doctor is going somewhere with holo emitters they don't need to be transported and can just be data transferred.
Personal rant follows:
!The whole hologram plot in Voyager is honestly poorly thought out, and it basically feels like if you followed their logic Chat GPT would be a protected federation citizen. I get that the writers wanted to give the doctor legitimized personhood, but it feels like they forgot to think about what that would mean for literally every other hologram. !<
!Like, they give that one species holograms to hunt, does that mean they invented a species doomed to be reincarnated as prey forever? Is, Moriarty sentient, and if so is trapping him in a simulation moral? If they just run a hologram long enough does it gain sentience? How are they testing for this? Does that mean Vic Fontane is sentient even though he probably would say he isn't? What about that weird Irish bartender Janeway does - fair haven ran for a while, how long does it take? If you run a training program are you committing infanticide? Is turning off a hologram even moral?!<
What would be the molecular structure of a hologram? I would assume that the mobile emitter would be transported and once it realise that, when it was projecting the Doctor, it no longer does, it reactivates him.