I can be wrong, from the top of my head I can not think of anywhere where leaving out egy would be more natural than actually including it. (Fluent bilingual here.)
But yeah don't EVER try to count the 10,000th occasion, haha. That would be a tiny bit silly. I think it's just safer to go with including egy all the time and being corrected than leaving it out too much and sounding very unnatural.
Oh, but there are plenty of occasions. Professions, just to name one or two:
I am a doctor. - Orvos vagyok.
I am a teacher. - Tanár vagyok.
This is not a secret. - Ez nem titok.
My dog is a vizsla. - A kutyám vizsla.
This was a good movie. - Ez jó film volt.
Etc. In some of these situations, Hungarian prefers not to use the article, in others it is optional or usually used. I don't know if there is an actual rule, or guideline, the best way may be listening to the natives.
Not necessarily. My observation is that if you say that something is a something else then you don't need 'egy' but that you do once you start introducing an adjective before the something else. Analogous with French eg, 'elle est actrice' but 'elle est UNE jolie actrice', but with a wider scope. But I may be wrong.
I think it has more to do with how much you want to quantify the noun. Egy is "one", too, after all (just like in French afaik). Sometimes it feels unnecessary to quantify the noun. I'd say it's generally more likely to implicitly set the amount when introducing something new. On the other hand, you may not want to "count" people, for example.
Omitting egy doesn't really make a difference, Ez nagy folyó still means This is a large river. Though This is a large river and This river is large (and similarly, Ez (egy) nagy folyó and Ez a folyó nagy) basically mean the same thing, there's an important grammatical difference, do it'd be wrong to mix them up.
How do you say "It is a big river" vs. "This is a big river"? I have seen the "Ez egy nagy folyó" construction used with both meanings in other lessons (talking about hotels, etc.). Sometimes the phrase is "Ez nagy folyó" and it has also meant either "this is a ..." or "it is a ..." Any rule here? Is it all contextual?
Yep, as Ryagon said, ez/az are dominant at this topic... although I wouldn't say Hungarian doesn't use personal pronouns for inanimate objects at all. Most personal pronouns are fine to use in the "coda", some place after the verb. Ő/őt are the universally uncommon ones for "it" nominative and accusative respectively.
It's contextual, more or less. The main issue is that Hungarian doesn't use personal pronouns for inanimate objects, and since this type of sentence doesn't use a verb either, you'd only be left with "Egy nagy folyó" as a literal translation for "It is a big river". That doesn't mean much.
So in Hungarian you're more inclined to use ez or az for this kind of sentence than using "this" or "that" in English.