Charles,
The major/minor sequence is the same for sixths when you think about what chords the shape is taken out of. Your confusion may arise because the sixths are generally voiced from the higher pitched note, eg. the first sixth in the key of G will be from the G on the 1st string with the B on the 4th fret, 3rd string. You can see that this fits right inside the G major chord but the interval is a minor 6th, when viewed from the perspective of the B to the G note. The interval would be a major 3rd when looked at from the perspective of the G to the B note. This is called the “rule of nines”, an interval and it’s inverted interval will sum to 9, ie. perfect 4th and perfect fifth, major 2nd and minor 7, minor third and major 6th.
So, in our G major chord sequence, the G major chord has a minor 6th, the A minor chord has a major 6th, etc. It’s confused me in the past , also.
Regarding your second question, you can play mostly in the key of G but the C minor and B7 chords are not in the key, so you might make an adjustment there.
John

Hi Charles,
the following two videos may help you:
Using Thirds & Sixths

Here’s an example using 3rds and 6ths, which I recorded yesterday:
3rds and 6ths example

Eric