There was only one Confederate States presidential election, and that was held in 1861. It was the first and only presidential election.

However, Congressional elections where held twice, and were held in odd numbered years, unlike US/Union Congressional elections which were held in even numbered years. The First Confederate Congress(as in the first regular term) elections were held on November 6, 1861, while the second was held in November 1863, however those elected in the second election where only able to serve for only one year of their two-year term seat(due to the defeat of the Confederacy). You also had Provisional Confederate States Congress, which came before the First Confederate Congress elections where held. The Provisional Confederate states Congress is also known as the Montgomery Convention, and that is where the formal beginning of the Confederacy started, and where the provisional government was organized.

As for out come of these elections, you can view the first elections out come here: http://www.csawardept.com/history/Congress/1st/ and the Second elections outcome here: http://www.csawardept.com/history/Congress/2nd/

I looked into this a couple of weeks ago. Jefferson Davis did not have to run for re-election, but there were two or three (depending on how one counts) elections for the Confederate Congress. The first one was for the Provisional Confederate Congress, a unicameral body, and others for the more conventional Congress. The short version (many more details in the older post) is that where military realities did not interfere (and occasionally where one would think they did), elections usually went on more or less as normal. Where Union armies or raiding intervened, things got complicated.

The First Confederate Congress had 35 (33%) of its House members carry over from the Provisional Congress. The real rate is slightly less since the 33% includes Missouri's appointments, while everywhere else at least had a kind of election. Half the Senate also came over, though of course they weren't senators in the Provisional Congress. Florida, Texas, and Mississippi voted their entire delegations out in favor of new ones. Only Louisiana and Virginia returned more than half of the provisionals to office. Members of the Provisional Congress were usually chosen by secession conventions rather than ordinary elections, which probably accounts for some of the difference.

It looks like I didn't go into incumbency in the Second Confederate Congress with my old answer, which concerned election methods more than results, so I'll do so now. Martis has 53.8% of incumbents returning in the Second House. Occupied districts returned their incumbents at a higher rate, 66.7%, vs 42.6% for the unoccupied districts. Fifty incumbents in total didn't retain their seats. The biggest turnovers were Georgia (one guy of ten retained his seat), Missouri (which arranged its first real elections of the war with an all-Confederacy canvass of Missourians and was rewarded by having all of two incumbents kept in office) and North Carolina (3 of 10). Every other Confederate state managed 50%, but Virginia, Texas, and Florida all tied at exactly that. In the Senate, things are a bit better. Martis has 18 Senators (69.2%) returning and 8 (30.8%) new. Kentucky, Tennessee, South Carolina, Florida, Arkansas, Louisiana, and Texas all returned both senators. Missouri and Alabama kicked both incumbents out. The other three states split, returning one incumbent and election one replacement.

Source: The Historical Atlas of the Congresses of the Confederate States of America:1861-1865 by Kenneth C. Martis.