For an excellent visual reference that might assist in answering the question, see Wikipedia's timeline.

To answer the easiest part of your question (the math part), since I don't know much about the opinions part, I created a spreadsheet. Excel, by the way, doesn't natively recognize dates prior to 1/0/1900 (not a typo). So that was fun.

Anyway, Martin Van Buren (#8) lived to see more presidents after his term (8) than any other president so far, although he may have gotten lucky with a couple short-termers: William Henry Harrison (#9) and Zachary Taylor (#12). This was, of course, a pretty tumultuous time in the nation's history, and it would be interesting to hear what he had to say about the lead-up to the Civil War. He died during the Lincoln administration.

In second place is John Tyler (#10), who lived to see 6 more presidents after his term was up. He also died during the Lincoln administration. He was from Virginia, and Van Buren was from New York, so they might have had some competing views on the Civil War. Again, I'll leave that up to someone else to answer.

(Edit: the stretch of presidents between Van Buren and Buchanan, Lincoln's predecessor, is also the longest stretch of single administration presidencies in US history - I say "single administration" to count a VP taking over after the death/resignation of a President as a single term. So this would explain why these two are at the top more than their longevity.)

Tied for third place with 5 each are John Quincy Adams (died during Polk), Millard Fillmore (died during Grant), Herbert Hoover (died during LBJ - might be interesting!), Nixon (died during Clinton), Ford (died during GWB), and Carter (still going strong).

So who has lived the longest after their term? Jimmy Carter is winning that contest easily, at 12,818 days. Second place is Herbert Hoover, who lived 11,553 days after the end of his term. Gerald Ford is the only other one over 10,000 days at 10,932. And you have to go back to John Adams for fourth place. He lived 9,253 days after the end of his term.

Of course, 8 presidents have died on the final day of their term, since they died in office. The shortest post-presidency besides those 8 was James K. Polk, who only lived another 103 days after the end of his term.

To answer the other part of your question, as I understand it: the largest gap in ages between consecutive presidents is between Eisenhower and JFK. JFK was 26 years younger than Eisenhower.

George Bush the Elder and Bill Clinton come in second place, with 24 years between them.

Th largest gap in ages of two non-consecutive presidents living at the same time is between Herbert Hoover and JFK. Hoover was 43 years older than JFK, and outlived him.

Next up is George Bush the Elder and Barack Obama who are 37 years apart. Carter is just a few months younger than Bush, and is 36 years older than Obama.

Interestingly, if Rubio or Cruz ended up being the next president, and either Bush or Carter survives until the inauguration, then that would take the cake for largest gap between the ages of two living presidents, at 46 or 47 years.