I promised to update this thread when I have done more experimenting with supplements. Better late than never.
For now almost 3 years, I have daily taken 2 x 1000mg pure MSM and 3 x 500mg green-lipped mussel capsules. I took care to choose a brand on the green-lipped capsules which has very little iodine so the regular iodine intake would be irrelevant.
Well, what can I say. MSM clearly had a positive effect on my skin, so I kept taking MSM. I believe the green-lipped mussels do help my joints in a way that I can quicker recoup from a bad joint cracking day.
Both supplements are very affordable and do more good than harm so I kept taking them till this day.