To your first question: Theodore Roosevelt attempted to run for a third term under the banner of the Progressive Party in 1912, after he fought for and lost the Republican nomination to Taft...but keep in mind, he became president following the assassination of William McKinley.

Under the 22nd amendment that first term would've counted toward his eligibility to run...but while he nearly served two full terms, it's not clear whether people legitimately believed he violating precedent at the time.

To your second question: not that I'm aware of. Post Andrew Jackson and prior to the 22nd amendment, there was only four presidents who served a full two terms

• Ulysses Grant, who left office plagued by scandals in his administration

• Grover Cleveland, who served two non-consecutive terms

• Woodrow Wilson, who had a stroke and was nearly incapacitated his final months in office

• Franklin Roosevelt, who served a full three terms and part of a fourth, and was to some extent the reason for the 22nd amendment