This is essentially a partial answer, but you're asking two questions so maybe someone else can provide the other half. I can only reply to your first question:

It seems to be exceedingly rare, if it occurs at all. The only other possible occurrence I could think of, and I wouldn't have thought about it if you hadn't mentioned ăgo > ēgi, is ĕmere > ēmi. But this train of thought is compromised by such cases as facere > feci, where a similar phenomenon occurs in Inlaut. You're basically stuck looking for irregular root forms (without prefixes !) of verbs that start with a vowel, and I am inclined to conclude that your initial premise of ăgo > ēgi as an augmented root is essentially incorrect.

One reason why I can see e- augmentation not staying a language feature in Latin, if it was inherited from PIE, is the verbal prefix ex-, which becomes e- before most consonants. This would introduce huge ambiguity between a lexical shift in meaning, and a grammatical one. I don't know any other languages from antiquity, so I'm not sure whether there is a similar prefix in other languages that might invalidate this point.

Since I don't believe there is augmentation in Latin, part of your last question is moot in this answer. But I can't seriously reply to the rest of it because I don't remember any rules why some words would get reduplication and why some wouldn't.

This is essentially a partial answer, but you're asking two questions so maybe someone else can provide the other half. I can only reply to your first question:

It seems to be exceedingly rare, if it occurs at all. The only other possible occurrence I could think of, and I wouldn't have thought about it if you hadn't mentioned ăgo > ēgi, is ĕmere > ēmi. But this train of thought is compromised by such cases as făcere > fēci or vĕnire > vēni, where a similar phenomenon occurs in Inlaut. You're basically stuck looking for irregular root forms (without prefixes !) of verbs that start with a vowel, and I am inclined to conclude that your initial premise of ăgo > ēgi as an augmented root is essentially incorrect.

One reason why I can see e- augmentation not staying a language feature in Latin, if it was inherited from PIE, is the verbal prefix ex-, which becomes e- before most consonants. This would introduce huge ambiguity between a lexical shift in meaning, and a grammatical one. I don't know any other languages from antiquity, so I'm not sure whether there is a similar prefix in other languages that might invalidate this point.

Since I don't believe there is augmentation in Latin, part of your last question is moot in this answer. But I can't seriously reply to the rest of it because I don't remember any rules why some words would get reduplication and why some wouldn't.