Can I split a single nonmetallic cable to multiple lights?
NOPE across the board. 1) Any exposed wire must be protected by conduit, enclosure or it's own heavy duty armor, as in only proper Armored Cable can be used. 2) You may pigtail or split the Romex only within the confines of the ceiling box itself using a wire nut or similar approved connector &amp; the box must then be securely &amp; entirely closed.

Basically, you need to make &amp; wire your own actual complete fixture. You wire it how you want &amp; have just single leads attached to each member of the Romex, just like any normal fixture. If the fixture's enclosure is metallic or conductive, it needs to be grounded along with or in conjunction with the sockets. I suggest finding a cheap off the shelf fixture &amp; adapting it to your needs.