I am definitely not an expert in history, which usually means I cannot answer any questions on this subreddit. This question however can be answered by manipulating data in one's favorite programming language.

So this is exactly what I did. I took the relevant dates from Wikipedia, and used python to find the answer to your question. I know Wikipedia is hardly a legitimate source, but I think it is trustworthy enough as far as dates are concerned.

So, there were actually three times in history when the US had six living presidents:

• When Abraham Lincoln became president, the following five presidents were also alive: Martin Van Buren, John Tyler, Millard Fillmore, Franklin Pierce, James Buchanan.
• When Bill Clinton became president, the presidents Richard Nixon, Gerald Ford, Jimmy Carter, Ronald Reagan and George H.W. Bush were all alive.
• When George W. Bush became president all of the presidents from the previous list except Richard Nixon were alive, so the count remained at six.

For the bonus question, you already knew that John Adams was the only living president after the death of George Washington, but he is not the only example. Ulysses S. Grant was the only living president after the death of Andrew Johnson, Theodore Roosevelt was the only living president after the death of Grover Cleveland and Richard Nixon had the honor after the death of Lyndon Johnson. Herbert Hoover was actually the only living president twice: once after the death of Calvin Coolidge, and once after the death of Franklin D. Roosevelt.

Of course, all of this information is conditional on me not having any bugs in my code. If someone finds a mistake please let me know.