Escribí הסוס שלך יפה y como no puse las vocales, no me la dio correcta :( Me la dio "casi correcta"...
That is possibly technically correct, but it sounds somewhat awkward and unnatural in English. I'm not certain either way if it's grammatically sound, but it's not a construction most native speakers would use to express the meaning, and I'm not surprised it wasn't in their initial list of acceptable translations.
No - a thing that is owned by a person is inherently definite and needs the article.
The only exceptions I'm aware of are אבא and אמא, which are taken to be definite without the definite article; I've heard various explanations for this, from "You only have one of each so they must be definite" to (more likely IMO) the fact that the words were formed from אב and אם by adding an Aramaic definite article to the end, and thus effectively have the definiteness baked in.
In any case, you need the ה. I don't know if there are other exceptions (see disclaimer above), but for the most part, if you assume that a thing that is owned is definite and thus requires the definite article, it's a pretty good rule of thumb.
I'm a native speaker, and I think you're correct with the rule. My only comment is is that other family relatives also don't need ה prefix to be yours: סבא, סבתא, דוד, דודה, אח, אחות. You'll notice that some of them are not from Aramic, and none of them has to be unique. It's a mistery to me, too; possibly they lost the necessary ה by analogy from אבא and אמא (which in turn lost their ה for one of the reasons you quote, or another reason; FWIW I find the "there is only one" a more likely reason than Aramic etymology, though if we knew the history of using these words in modern Hebrew it would help theorizing). Note that the relatives pattern stops with the above: בת, נכדה, כלה and חמות, and their masculine counterparts, do require ה- if they are שלי.
Interesting. You're right, it's hard to specify the differences even for us natives.
The main difference is that in my examples I used the definite article, which made הוא or זה unnecessary.
As he also mentions in the end, ה changes things a little bit. Anyways, I disagree with him. He said זבוב חרק isn't a sentence (which is true), but why is זבוב מהיר? It is a grammatical set of words (a fast fly) but it also can't stand by itself.