How many notes does it take to state the key? To Have "Tonality"?
Tonality is the theory that describes what it is to be in a key. Thus being "in a key" and "Tonality" are more or less synonymous.
The "how many notes" issue is the sort of question that music theorists debate without necessarily coming to a definitive answer. The video presents two models of tonality and clearly makes the case that the video author prefers the Yavorsky model.
The Yavorsky model presents a four-pitch minimum, and the Schenkerian model (as presented here) requires six. Thus, within the context of the video, it takes four pitches to establish Tonality/a key.
How many notes in a "melody" do you need to state the key?
Here, too, there is ambiguity, and it depends on the restrictions placed on "melody". In the most strict sense, a melody is not sufficient to define tonality. Establishing a key requires more context than single notes presented sequentially.
However, there is an idea of "compound melody" in which a single melodic line functions as though there were two separate melodies intertwined. In that case, it would again take four notes, according to the Yavorskian model.
Do the ideas of root progressions and the fundamental bass coincide with what I'm inquiring?
Root progressions and fundamental bass are both concepts that can be understood independently of tonality. However, in the Schenkerian model as shown in the video, we see a root progression of I-V-I.
Fundamental bass is a pre-tonality concept that proved an important idea as the theory of tonality developed. But the core of "fundmental bass" is just the idea that, given several simultaneous pitches, there is one pitch that serves as what today we would call the "root" of the chord.
And can a key also be stated in a ONE chord (or does it take two?)
At least two chords must be present, because given only a single chord, there's no way to establish that it is a
I chord in the sense of Tonality.
Also, I'm having trouble understanding the fundamental structure. Is it the fundamental line or the bass arpeggiation that states the key. OR, is it something about both that supposedly works together to state the key?
It's the two notes in each of the two chords, working together, that these theories claim defines Tonality.
And so lastly, which theory is true or false. "The Yavorsky Model" or "The Ursatz" (Schenkers Model)?
Neither is true or false. They are two models to describe a certain musical vocabulary. One can choose the theory that best reflects one's experience of the music or that best illuminates a particular aspect of the music.