I cannot answer your first question, but I can answer the second one. Yes, it was discussed. It had been advocated by President Andrew Jackson during both his terms in office, a result of his feelings after having lost the 1824 election in the House of Representatives.

In several of Jackson's state of the union speeches, he advocated for a single term of four or six years for the President, with no ability to run for re-election. When that did not happen, he ran for a second term, but still advocated for the term limit. In keeping the informal tradition, he declined to run for a third term and ended his presidency after eight years.

This earlier post of mine fills in the details of Jackson's plan and what came of it (not much), along with sources.

As to why the Confederates adopted the same thing, I am not sure, so hopefully someone with more knowledge on that aspect of history can fill you in.