Action and Intonation
There should be a slightly higher action on the bass side, but 1/4 inch sounds excessive.
A very high action will cause intonation problems, this should be obvious because in depressing the string you are stretching it (or “bending” it, if you prefer) – the higher the action the greater the stretch.
However, if the action is high on low-E, it’ll be just about as high on A, if A intones correctly then a dud string is indicated.
Tuning
Tuning the guitar from the high end might be unusual, but is not in itself bad practice, however, tuning from high to low, or, from low to high, can allow errors between strings to “multply” – a small, “acceptable” error between low-E and A when repeated across all the strings (A-D, D-G, etc) can become unacceptable.
If you do not have a great ear (that’s most of us), it is good practice to tune distant strings. The following string order is taken from The Link I Cannot Find.
Somehow tune low-E to pitch.
Tune 2nd-fret D to 12th-fret harmonic on low-E
Tune 3rd-fret B to 12th-fret harmonic on D.
Tune high-E to low-E (open, or, 5th-fret/12th fret harmonics)
Tune G to 3rd-fret low-E
Tune A to 2nd-fret G or 5th-fret high-E
It is not a bad idea to double check as you go along, so when tuning the G, say, it is worth checking against the 5th-fret D, etc.
There are two chord shapes that serve as a shibboleth, if these don’t both sound equally “right”, you ain’t in tune:
“E5” 022450
and
“A5” 002255
Tedster’s Overtone Compensation
The overtones of stringed instruments tend to be a little sharp, this phenomenon is more exaggerated in heavier strings. HOWEVER, this effect is small on all instruments except the piano, if you’re flattening your low-E any more than the bearest creak of the tuning peg you are not compensating for overtone sharpness but some other “fault” in the guitar.