I’ve been painting for quite some time now-over ten years, I think.
Are you confused by the "quite some time" part? It is just a saying that means "for a long time". The sentence has the same meaning as "I’ve been painting for a long time now. Over ten years, I think."
I try to go to an art museum once every other week-so that’s twice a month.
"Every other week" means not every week, but one time in every two weeks. One week to an art museum, the next week not, the next week to an art museum, the next week not...
For most his faults, I think Picasso was a great painter.
This means "even though he has some flaws, I think Picasso was a great painter".
Let us know if any of this is still unclear and someone (maybe me) can try and explain better. :)