I am not sure if I have misunderstood your question or others have.
The idea I got from your posts was, which is the third, which is the fifth and which is the first when you play a chord.
This is confusing on a guitar and makes it harder to understand, on a piano or keyboard it is very very simple, since you play 1st - 3rd - 5th in that order.
On the guitar, most of the time you do not.
Take for instance the most simplest Major Barre Chord.
1
1
2
3
3
1
F Major. F A C
If you look at the notes, you do play the 1st (Root) but then the second note you play (if you were to strum downwards) is the 3rd fret of the 5th string.
Well that is C, thats the 5th. The next note is another F, a repeat of the root, and it isn't until you get down to the 2nd fret of the 3rd string that you finally hit an A, the 3rd.
So it is slightly more confusing on the guitar then perhaps it would be on a keyboard.
Also I don't believe you do need to memorize every note on every string to improvise well. Scales have shapes that move around, there are perhaps 7 Shapes for the major scale, 1st position, 2nd position etc etc.
These are movable, once you have learnt the shapes, you can move them around on the fretboard without really knowing the notes you are hitting.
For instance the Pentatonic Scale is a very very well known scale, it is used extensively in both Blues playing and Heavy Metal. It is also a very simplified version of the Major scale, and the Melodic Minor Scale is only a slight variation on that.
It is effectively, painting by numbers, rather then understanding the concepts of the music you are playing, but to truly understand the concepts is a LOT of work, if you are interested in playing Jazz for instance, then yeah, you will have to memorise the fretboard, you will have to understand just how a Sus4 +9 chord is created and how that changes the notes in the scale you are playing in, and you have to know the scales well enough to move with the chords most of the time when they change.
Extremely difficult to do, which is why Jazz players are soo revered.
If however you just want to play a bit bluesy, some heavy metal licks or 'general' improvisation, just memorize the scale positions.
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