Learn English – Use of article in front of product names

Your description of the use of the definite article as opposed to the indefinite article is accurate. The presence or absence of one of these articles, however, in no way alters that standard usage.

As for why the article may disappear, it strikes me that when the article is there, it draws attention towards the adjective and away from the person. Let me use sentences 4 and 5 to illustrate:

It seemed Walter didn't pay any attention to a tearful Kitty.

Stated this way, the sentence indicates that Walter is not paying attention to Kitty's tearfulness; he is not moved or affected by her distress.

She hasn't got lovely eyelashes like marvelous Monica.

In this case, the attention of the sentence is squarely on Monica, and "marvelous" is relegated to being a pure adjective, neither calling for nor receiving any unusual or special notice from the speaker. If we added "the," it would put the attention on the fact that Monica is marvelous. (This can, by the way, be a vehicle for sarcasm, if you want to point out that she's really not all that marvelous after all.)

Is the difference simply that, general v. specific?

No, but it is the starting point for teaching English as a foreign language.

I suggest starting out with only general vs. specific usage, and then adding other rules bit by bit (giving precedence to those most used):

1. general vs. specific
2. always before ordinal numbers (but never with possessives)
3. always before superlatives (but never with possessives)
4. always before river names etc.

The point is to avoid drowning the learner in rules, thus only giving more rules when the previous are understood and assimilated. I recommend getting a list of rules from a good grammar book and choosing the ones more relevant for the learner's level.

Omitting article “the” in front of plural nouns

It seems I missed the 'question' in the title and answered only the last one.

'The' isn't used with plurals when that plural implies a general reference, only when that plural implies a specific group.

So...

Students learn better when teachers are passionate about teaching.

is the equivalent to: "Any student learns better when any teacher is passionate about teaching."

Whereas...

The students learn better when the teachers are passionate about teaching.

implies that you're not talking about students in general but a particular group, for example those who study in a certain school or at a certain level; and the same goes for the teachers: you don't mean teachers in general.

Conclusion: Without a context that clarifies whether the sentence relates to a specific group of students/teachers, the first sentence (zero article) is preferred.