Notes within a mode aren't limited to a certain octave. Take your C, F, G progression. The tonal center is C (as the G chord pulls toward the C and the progression would nicely end on the C chord). The appropriate mode is C Ionian (C, D, E, F, G, A, B). One way you can get this from taking the noets of each chord C, E, G, F, A, C, G, B, D. Now order them, eliminating doubles, starting on C, since that's the tonal center. C, D, E, F, G, A, B, that's C Ionian.
Now that you've got your mode, using it... Loop the C, F, G progression and play over it usnig the notes from C Ionian. You can use any C, D, E, F, G, A, or B on the fretboard, regardless of the octave its in. The essence of a mode lies within the harmony. As you can see, this boils down to simply playing in C Major over a C Major chord progression.
You're talking about inverting chords - that has to do with voice leading to get the most pleasing sound or easiest fingering for chords. For example, it sounds kind of weird to play a bar-8 C Major, then a bar-1 F Major, then a bar-15 G Major. It makes more sense to have the chords close to each other on the fretboard, such as open position C Major, bar-1 F Major, open position G Major. Notice that the C chord is in 1st inversion, since it has the 3rd of the chord (E) on top. This sounds good because the next melody note is an F in F Major, then it goes to the G in G Major. Small movements in the melody notes between chords are usually the most economic and also *usually* sound best.
"You must stab him in the heart with the Bone Saber of Zumacalis... well, you could stab him in the head or the lungs, too... and the saber, it probably doesn't have to be bone, just anything sharp lying around the house... you could poke him with a pillow and kill him."
- Aqua Teen Hunger Force, The Universal Re-Monster