crotch check
The doors have a setting, "manual" or "automatic". When in automatic, as they should be during the flight, if the door opens, the slide inflates. Emergency exit doors are always in automatic, as they're only used for emergency exit. On an A320 there's 4 of these over the wing. An A319 has 2, an A321 doesn't have any -- all doors can be used as normal doors.
However when you land, that's a bad thing, as the jetty driver would be in for a bit of a shock! As such, before opening the doors they need to be set to manual.
The reason you're told about the red dangly cord to inflate the slide is in case this hasn't happened (or the auto-mechanism failed)
Cross check means that someone else checks the setting is at the right location (i.e. crew member 1 does door-left, crew 2 does door-right, then crew 1 checks door-right and crew 2 checks door-left). On some flights this means physically looking at the indicator, on other flights it seems a quick nod to the counterpart.
P.S. is there any reason other than convention that people almost always board from the left?