Whether you add egy in these kinds of sentences or not is totally up to you.
Not exactly. Egy = a/an ; therefore whenever you'd say a or an in English, you should say egy in about 99.99% of the time.
Leaving it out of only one in 10,000 occasions feels a bit extreme. ._.
Three months of learning later I'll have to agree that the inclusion of egy sounds much better in most cases; it gives a little more structure to the sentences. But it is not needed per se.
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.
If you left out egy, would it mean "This river is large," or would you need van?
No, you wouldn't need van, but it would still be different: Ez a folyó nagy.
In the structure this/these/that/those [noun] the definite article has to be included.
If we left out 'egy', it would mean 'This river is large', yes. The predicative is an adjective, which characterizes a state of being - and since we say it in 3rd person, the verb 'van' is omitted. (The omission is compulsory.)
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.
'gy' occurs in both 'egy' and 'nagy'. is it sounded in both? or is it sounded only in 'nagy' because it ('nagy') starts with 'n'?
All the letters in Hungarian each should be pronounced the same, regardless of their position. The 'gy' of egy mingles a bit with the following nasal, so it doesn't sound as clear, but it's still there.
So, is 'ez' this or that? I think it translated to 'that' in one of previous lessons.
Ez is 'this', referring to something which is in close proximity. In some special constructions you can also translate it as 'that' in English, but mostly it should be 'this'. 'That' is az.
Feel free to report anything that's incosistent with this.