My question is a) How do I bring this disconnect between work and
after work to this person's attention/ HR
What this person does or doesn't do after work isn't any of your business. What this person posts on Facebook is not your business, either. What this person does on the job is.
Instead of focusing your attention on a "disconnect", focus on work, and how this person's sick days are affecting the assigned work (if they actually are).
Something like: "You know, Bob, when you are out sick others have to pick up the slack. Taking so many sick days really puts a burden on the entire team. While you can't control when you get sick, I hope you try harder to be in work when you can, and stay away only when it's really necessary."
b) Should I even treat this as a medical condition and request
doctor's note etc... or should I treat it as performance issues and
remove the person
Most companies have a specific policy about this. Consult HR and ask.
If the absences require a doctor's note per HR, then bring it up with this person right away, and require one.
If there is a performance issue to the extent that you feel this person must be removed, then clearly you haven't been handling your managerial role in a timely manner. Again, consult HR for guidance and find out if a Performance Plan is in order, or if you should deal with the poor performance some other way.
c) Will treating this as performance issue open us to legal action due
to medical condition etc...
Maybe. It depends on many factors - your locale, the person's actual medical condition, if there actually is a performance issue, etc.
Again, you seem unclear. Consult HR for advice and guidance.
You post says nothing about poor performance - only that you feel wronged by someone who has unlimited sick days, calling in sick but doing non-sick things after hours and weekends. Try to get clear regarding what is the real problem here before acting on it.
[Edit] Sounds like the problem you should concentrate on is the poor performance, and not the other factors that may or may not contribute.