First of all are their names: They both have second names on their ID cards but they did not put them onto the booking.Is this going to be a problem when it comes to checking in at the airport?
I read all the warnings on websites about this. Then I accidentally showed them the wrong passport (I have two) and they waved me through as normal.
They have paid for checking in their luggage online. Do they have to check themselves in online on the day of flight before they get to the airport
Yes - absolutely. Otherwise it will be expensive. But don't worry about the "day of the flight" thing - their online check in opens long before that.
They have one print out with both their names, do they have to print two- one for each?
For boarding pass yes (and print spares) for the e-ticket, worth doing just in case.