Did she study at any establishment in particular?

"became involved romantically" might work better here

If you have room, I would just suggest adding how/why he returned to Paris because the bio goes straight from his refuge in Spain to him dying in his Paris home. Something as simple as "Picabia eventually returned to his home country..." would help chronologically.

"beginning" instead of "causing" perhaps

"a very new and abstract concept" may work here as well?

"not being a household name" would suffice

obsessed is a strong word, it works but if you use this word I as a reader would want you to elaborate further on how he was obsessed with it (versus perhaps fascinated, for example)

"and" would probably work better here

America / the United States

I would suggest using "beginning with" or "starting with" rather than "starting during" ; it just makes the sentence flow better.