Garage subpanel wiring
Looking at your sub it was done correctly , the white neutral bond was not connected so you can connect the 14-50 , you will need to remove one of the double pole breakers and yes you use all 4 wires 2 from your new 50 amp breaker 1 ground and 1 neutral. , great question and good photos. 
Make sure to get the correct brand and type of breaker for the sub I can’t quite make out the brand but the main is a Murray , but the tandem breaker is Bryant this is something I have seen many times and usually not code compliant for 2 reasons , Bryant breakers were not listed for a Murray panel even though they fit. Item 2 is early panels were limited on the total number of spaces so manufacturers came up with “cheaters” early tandem breakers that did not have a rejection feature allowing the buss to be overloaded, these breakers usually have a sticker stating they are Non Current limiting. (Just want you to know).
If you can find the proper double pole breaker for the sub all may be fine but I would want the main panel updated so there is not a problem if there is a fire later even though everything on the sub is correct there could be problems because of the alien breakers in the main.