Electrical – Can the sheath of nonmetallic cable be removed prior to entering electrical metallic tubing (EMT)

What size box is it? It should either have a volume printed on it, or will be a standard size. Without knowing the box volume, we can't give you a definitive answer.

Assuming it's all 12 AWG, it's at least (12 (current carrying conductors) + 1 (grounds) + 4 (device fill)) * 2.25 = 38.25 cu. in. currently.

Adding another two conductor cable with ground would require an additional 4.5 cu. in. For a total of 42.75 cu. in., not counting any clamp fill not mentioned.

As long as the box is larger than this, you'll be okay.

Note that if you put the third device in, you'll have to add another 4.5 cu. in.

See this answer for more detail on box fill calculations.

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 & the box must then be securely & entirely closed.

Basically, you need to make & wire your own actual complete fixture. You wire it how you want & 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 & adapting it to your needs.