If your singing a third above the melody and the last note in a phrase is the highest note in the chord, e.g. chord is A and melody note is E, what should you sing as harmony?
Depends what style you're thinking in. In 'Harmony 101' G♯ would definitely be considered an 'avoid note' over an A major chord. But in more modern styles a maj7 chord is hardly considered dissonant at all, continuing a run of 3rds might be a good enough reason to allow it.
Example A is perfectly acceptable in jazz-infulenced styles of the last 100 years or so. (Or Debussy...) Bach would more likely have preferred B, ducking from '3rds over' to '3rds under' when they started producing naughty things like added 6th and maj7 chords.
Play them both, absorb their sound, choose which is right for YOUR music.
(P.S. I was transcribing some Everly Brothers songs recently. The printed copies, if they show a vocal harmony at all, normally suggest nice tidy 3rd and 6th intervals. Listen to the tracks, you'll find the boys weren't afraid of the odd parallel 5th! And it doesn't sound 'bare' at all.)
chord is A and melody note is E, what should you sing as harmony?
It depends on what kind of harmonic style you want. G# will make the chord an Amaj7, which might be too jazzy for some styles of music. An A will be a fourth, which isn't as pretty as a third. A C# will make it a sixth above, but at least it's pretty and not too jazzy.
Harmony singing is usually a third above the basic note - but - it will depend on what the rest of the bar is doing.
If the bar is V (in key A) the the note above E will be G♯, but if the bar is I, then G♯ won;t usually sound that good. So a 4th may be a better harmony. A and E. It will sound not as harmonious but at least it will fit. Not really a situation where one harmony in thirds will work.