#1
So I was wondering, since I've been looking at harmonization lately and it makes sense. Like the lower note seems to dictate how the higher note will harmonize. Like a bass plays an A and then a guitar plays a C# then it's a major third harmonization right?

Well see it all makes perfect sense but then I look at inverted chords and think "wait, how come a 1st inversion A major chord not C# whatever?" So yeah, can anyone clear up how inverted chords work a little more clearly for me?
#2
the easiest way to put it is that the chord implied can usually be interpreted by the progression being used. say we know that its a i-iv-?-VI progression and spelled out low to high the chords are (with the first note in the bass cleff)

E,G,B,E - i
A,C,E,C - iv
D,B,F#,B - ?
C,E,G,C - VI

the ? indiciates our missing movement, is it a D with an added 6th? well you might be able to look at it like that, although the most likely answer here would be a B minor inverted with the bass in D. how i came to it is this, i know the i,iv,VI makes up part of the E minor scale by looking at the notes and knowing that the Em is the i. the ? or missing part is the V, a i,iv,v,VI movement is somewhat common in modern rock music and would make more sense than a i,iv,VIIadd6,VI harmonically, also the v -> VI follows the forward motion created by the i-iv in a consonance manner more so than the i,iv,VIIadd6 ALSO the v resolves to the VI in a more consonant manner than the VIIadd6 -> VI. however the second time through you could highlight the D note perhaps with additional harmony and consider it a i,iv,VIIadd6,VI which will have the VI push back to the i to continue the forward motion. theoretically if i were to use both i would use i,iv,VIIadd6,VI then i,iv,v,VI as the VI resolves to the i after that movement and sounds as though the song is at rest.
#4
it can yes, also its easy tell an inverted chord by the overall tonality and where the bass note is located, say for example the bass note as listed in the ? above is a D but there are a bunch B's in the treble clef, i would probably still call that a B, say however the bass cleff is a B but there are several D's in the treble clef, i MAY call that a D, considering how the movement of sound is. all in all, singuarly on a simple 3 note chord you can imply any bass note you prefer when writing something in the treble cleff, say i was to look at just a treble clef that i was writing and it has

E,A,C

in the key of E minor (btw one of my fave scales, its why i use it so much)
i could call this

Am
E4add6
or C6 (i believe, i would consider it an inverted version of C,E,A)\

and depending on which bass note i used and what chord it was going from and to i would give it a different name, if i was going from Em to C i would consider this an A, if it was between E and D, such as E,?,D i would probably consider this a C.

one way to tell if you're just missing the name of the chord is to look at the motion of the notes in the treble and bass and you can usually make an educated guess once you know your theory a bit. i recommend reading up on "figured bass" as it can help in determining things like this. i'm kinda surprised no mods jumped on this one.