¿llevas viviendo mucho en Barcelona?
 

I it means have you been living in Barcelona for a long time, but it sounds unusual to me. I would say ¿Cuánto tiempo llevas viviendo en Barcelona?
Is this usage that sounds strange to me used in Latin America?

The word order is what sounds strange to me, but it's true that sometimes the speaker thinks faster than they speak. Also, we often take advantage of how flexible Spanish can be.

I would have asked: "¿Llevas mucho (tiempo) viviendo en Barcelona?" (Have you been living long in Barcelona?)

This question is focusing on appreciating the time the other person has been living in the city. By asking this, the speaker might be implicitly interested in how adapted that person is to the place. When you ask "¿Cuánto llevas viviendo en Barcelona?", even if the idea is basically the same, you express a "more neutral" interest. ;)

Since all three questions will get the same answer, they're synonymous to me. :D