It is difficult to sort out what may be going on based on the information thus far. I will advise to just throw out the opening observation of a kicked out virgin. At this point it is irrelevant. I will also advise to throw out the dead queen in cage observation. Also irrelevant at this time. Those are past, by-gones. What you need to do is thoroughly inspect the brood nest to determine what is going on right now.
With respect to your questions:
What should I do about the space issue?
- FFF = which means minimum Four Frames Free at all times. If you do not see FFF at inspection, add space.
Is there a queen?
- are there eggs? yes there is a queen
- are there no eggs? no there is not a queen
- are there queen cups?
- are there queen cells?
- is there evidence of past queen cell(s) that emerged or torn down.
( egg laying interruptions due to a swarm, emergency self requeening, failed supercedure, beekeeper error/injury, ... are all set aside at moment)
How do I get them to draw out the super?
- You don't. They tell you they need a super by showing you that they do or they not have the needed FFF
Can I harvest honey?
- Is there FFF?
Add space or harvest honey to create FFF at all times
Should I do another split?
- at this point in the season the decision you are to make is if you want bees or if you want honey.
If you want honey, NO do not split. Put the bees to work
If you want bees, sure - you can split any time and as big or as small as you want.
- Are there eggs?
- Are you buying queens or leaving to the bees to raise one themselves?