To answer your questions:
1. It definitely does not sound like Meneire disease. While Meneire attacks are episodic, they aren't typically minutes apart, but rather days and weeks. Usually, hearing loss precedes the first attack of vertigo/dizziness. And tinnitus almost always accompanies it. It also tends to affect people in their 40's, though it can occur earlier. At this point, there's absolutely no reason to suspect Meneire disease.
2. The only safety issues with traveling will be the risks of flying itself. It will not cause anything unsafe medically related to your dizziness. The pressure changes that occur with flying could potentially exacerbate the dizziness/vertigo, but will not cause medical harm. (Note, being dizzy and off balance could cause harm if you fall; but flying itself won't cause medical harm in your situation.)
3. One can't predict whether you'll "never" have another acute attack of vertigo again; but in most cases, labarynthitis is a self-limited disorder: you have the attack, you get better, and most won't have recurrent attacks. (I assume by "disorientation" you are referring to dizziness/vertigo.) If it's labarynthitis, it typically subsides within a few days to a week or so.
All in all, Meneire disease is highly unlikely based on your description.
There's a lengthy list of things that can cause acute dizziness, but common things being common, the most likely cause if you took 1000 people your age with your symptoms would be labarynthitis. A few of the numerous things in the differential diagnosis would be eustachian tube dysfunction, benign paroxysmal positional vertigo (though most people with BPPV are over age 50), brainstem or cerebellar stroke (not very common in your age group), a cerebellopontine angle tumor (not common), a demyelinating (wearing away of nerve insulation) lesion such as multiple sclerosis, toxicity from various medications and/or illicit drugs, etc. With acute onset at your age (particularly if you can recall recently having even a mild upper respiratory tract infection, but that's not a requirement), labarynthitis is far and away the most common cause.
1. Wearing earplugs should not cause any problems.
2. Assuming this is labarynthitis (As I cannot evaluate you personally, I can only speak in the hypethetical sense.), when the dizziness/nausea resolve, resuming exercise (starting as tolerated) is fine, taking precautions to avoid falls.
3. What you are describing is not tinnitus. Tinnitus is a ringing--less like a telephone and more like cicadas (those noisy insects that appear every few years). The "pulsation" you describe could be a benign spasm of the muscles (very tiny) of the inner ear related to inflammation; or it could be from pressure changes associated with eustachian tube (a canal between your inner ear and the back of your throat) inflammation. It definitely is not tinnitus. Hearing loss (and it's pretty frank hearing loss--not alteration in what you hear or amplification of hearing) usually occurs prior to onset of dizziness, but less commonly can occur after. Meneire disease is not that common; and your case doesn't really sound like it at all.
I'm glad you're starting to feel better. I feel quite confident that this is not Meneire disease; but only time can really tell.