The season will end when patch 5.4 happens. Your cap will also reset, to stay in line with the new tier of gear. I imagine they will eventually implement the catch up system again for this pvp tier as well.
1) No official sources as to when the season ends yet, there will be a 2 week 'warning' which Blizzard will post on a tuesday soon™. As Garrosh testing has been completed-ish and patch notes are slowing I imagine it'll end in 3-4 weeks.
2) The catch up season uses this formula: (Current week of the season) * 1000 – (Conquest earned thus far this season).
This means that when a new season starts, your catch up will be completely wiped (0 * 1000) = 0.
Blizzard usually give a two week warning before the next major patch and end of season. Until that happens, we really don't know. It's getting close though.
Barring potential unintended behaviour through bugs, there's no way to carry anything from this season over to the next one (other than actual gear). Your conquest cap should be 1800/2200 the first week of s14; all of your current cap just disappears. Any conquest you've actually earned but not spent is converted to honor. The only useful thing you can do with that cap is earn it and buy gear. If you have any interest in pvp, I recommend that you do this as catchup conquest is easier to earn than honor.