It's not uncommon for melodies to include notes that aren't in the chord, especially on weak beats of the bar. The article linked is not really a good way to understand popular music or jazz music (and my common practice music theory isn't really good enough to give an appraisal of it's usefulness to analyse classical music in an explanatory way). The thing is 20th century popular music (especially post 1950s) has incorporated a whole new harmonic and melodic vocabulary that comes largely from the blues, but then also integrated that into the existing western harmonic framework, and so while more conventional "classical" explanations of chord functions are relevant in many cases, there is also a whole other set of vocabulary that completely breaks that down.
But the melody you linked contains nothing particularly weird (other than being in a horrible key to read). It's really normal in modern music to have melodies that have non-chord tones on a weak beat, and which resolves to chord tones. In classical terminology, a "passing note" has to resolve to the next closest chord tone, but in real music, melodies often jump around to a different chord tone. If you transcribe a bunch of pop/rock/blues/funk/soul melodies you will find this happens all the time. As a non-classical theorist it's easy to forget that this was ever considered unorthodox.
Let's answer your specifics before making a more general point though:
1) So the first bar goes 1 b7 5. That's a really really common melodic shape*, you'll hear that everywhere. You can think of it as forming part of a minor 7th arpeggio if you want (or dom 7 in a major 3rd context), but it's really not necessary to think of it like that. I guess that melodic pattern probably has it's origins in the blues, but now it's just been assimilated into pop vocabulary. Similarly you often find 5 b7 1 on the way up (think "my momma said to leave you alone" in "chain of fools" for a memorable example). So you can think of the b7 as a "blue note" if you want, and in a major key it really functions like that (just like the b5 or b3 also do in blues-influenced major key songs)
In a minor key though you can also think of it harmonically, as the 7th is also just valid chord extension for a resolved minor chord. In jazz music you'll hear even more "spicy" chord extensions being sung/played over minor chords, too, the 9th for example is a "clash" or "tension" in classical theory, but in jazz (and therefore by extension, sometimes in pop), the 9th is often just considered an "interesting colour" over a minor chord, and you can just leave it there to hang out. Because of the way this particulary b7 is integrated into the melody, I would interpret this as a "melodic note" rather than analysing it as a "chord extention", but in another context, you could equally hear a flat 7 (or a 9 even) being sung in a strong position over a minor chord, and there's nothing wrong with that.
https://instaud.io/3DFU singing 1s, 7ths, and 9ths over a tonic to show you what I mean.
3) that isn't a really an accurate transcription of the melody, it's really just a "fall", a low semi-spoken note. But the transcriber had a to choose a note so they went with a the tonic. I'd have probably written it as a C# if pressed to give it a note, but honestly it'd be best written with an X notehead with a fall after it.
4 & 5 ) What we have here is a "recontextualisation" of the original melody. You see this a lot in pop music, the same melody repeated over a new chord to give the notes new context, it's a useful device. Since none of the melodies notes clash too much with the new chord, it works well. As it happens, it's not a bad fit for the new chord anyway, the D# just becomes a weak tension that then resolves stepwise to the C#; a chord tone. But even if that wasn't the case, our ear is much more forgiving of non chord tones when they occur in a repeated melody from previously, because the "echo" of that melody is still in our head and so we sort of enjoy the tension between the familiar melody and its new context. This is particularly effective when the melody implies the tonic chord, because we then sort of hear the melody "pulling us home", by implying the home chord while another one is being played.
A good example of this the use of the "I wanna run away" and "You and I" melodies in the Galantis "Runaway", they serve to essentially "pull us home" to the tonic while the b6 and b7 chord are being blasted.
6) See 1), minor 7th is fair game as a chord extention in jazz, soul, R&B etc. That said, is does actually resolve up to the A#, so I don't see the issue. The drop to the 6 is just a bit of "ornamentation" on the 7th, just adds a bit of interest to give a bit of movement. You could equally sing the the F# as a G# or ornament the G# by going up to an A# instead of down to the F#. That said, I think it's better as the F#, the melodic shape is better with "you were mine" as a the highest note, it gives it more impact.
7 & 8) that little melodic figure just anticipates going to the tonic. It sort of implies like "F# G# A# (D#)" but you don't need to actually sing the D, it's kinda "implied". In classical terms it sort of functions like an appogiatura, or maybe a suspension, I forget which one is which, the one where a note from the previous chord is played before the chord gets there (I'll look this up and edit the answer in a minute ok).
Similar little riffs can do the same job. Here it was "F# G# A#", but it could have been (as straight 16ths so I don't have to write rhythm) "E# F# G# A#" "B A# G# F#" or "C# A# G# F#" or even "G#/A\G# F#" for a more bluesy feel (where the / indicates a bend, or microtonal inflection). It's just a little piece of "musical punctuation" that says "hey we're about to go back home".
As a more general endnote, because blues, rock, and soul music has been less formally studied and written about by academic music theorists, and because there's so much variety in how much different songs and musicians draw from each tradition, and because, unlike with classical musicians, most of the most innovative and significant musicians did their work outside the academy and so weren't writing or lecturing about it (and mostly weren't using the same musical language as classical musicians, a lot of the time didn't read or write music for example) there is much less of an accepted way to talk about it. Classical theorists sometimes have the annoying habit of simply saying "well pop music breaks the rules" and leaving it at that, which is essentially just giving up understanding most relevant music written after about 1920. And a lot of musicians/songwriters come at it from the angle of "what sounds good, is good, theory is for nerds, don't think about it too hard, use your ears", which I have a lot of respect for (well, especially the "use your ears" part), but if you're someone like me (and a fair few others on this stack) who likes to really understand how the music they like works, then you're going to have to come up with your own syncretic framework that kind of incorporates classical and jazz theory knowledge, but also incorporates your knowledge of how harmony and melody function in the songs that you like (which probably has to include the blues if you're really going to "get it" imo), and bits and pieces you've heard about and read about on the internet. And then you take that, and mix it into 1 big soup of "post 20th century music theory" I guess.
For an example, see this question here , analysing how bVII chords function in rock music. That is an example of functional harmony, (none of this "oh it's nonfunctional dw about it" nonsense) that nevertheless exists completely outside the classical or even jazz framework, but is a really really common device in rock and other popular music. Any working musician who plays a lot of repertoire (especially keys and guitarists) will have some ideas in their head of a sort of "practical music theory" that explains the music they encounter, but there's not yet a completely formal framework (as far as I know) that lumps it all together in a coherent and well-defined way. In a way, maybe that's for the best, since modern music is constantly changing and shifting all the time.
