Yes, some of them do.
Most of them did.
Byzantine art icons expressed the new visual language of the church. The artists used gold backgrounds to show the wealth of the bishops. This statement glorified the wealth of the church and many Byzantine churches were famous for the wealth of their art.
: icons.
Eastern Catholic Churches.
It is built in the Neo-Byzantine or Byzantine Revival style, imitating the churches and buildings built by the Byzantine Empire between the 5th and 11th centuries. It emerged in Europe in the mid 1800's, and peaked around the early 1900's. The Neo-Byzantine style has a heavy use of domes, windows, and circular shapes, as well as copper roofing and traditional brick walls.
Many large paintings hung on the walls. You are talking about "paintings", the subject. "Many large" are descriptive words of paintings. The paintings "hung"-- the word of the action for the paintings. Many large and on the walls--Those are the words confusing you. Step back from wordy or complicated sentences and try to simplify them that way.
The Byzantine Empire's most lasting contribution to the world's architecture were the churches.
Yes
Some were on rocks
The paintings they did on cave walls.
It depends what you like
Some byzantine churches are Orthodox. And some Orthodox churches are byzantine. Other than that, there's no difference. There are Roman Catholic churches which follow the byzantine form of worship, since they ceased being Orthodox in the 1400-1700s but retained their liturgical practices. And there are plenty of Orthodox parishes which have never followed byzantine norms for worship (Russian, for example, or in some places, Western Rite, especially when an entire Anglican or Roman Catholic congregation has become Orthodox). If you hear someone generically refer to "THE Byzantine Church," they are probably referring generically to the Orthodox Church, giving credence to its roots in Greek thought, culture, and language.