Is it OK to install Romex inside rigid or flexible conduit?

Generally yes, where the cable in the conduit and the conduit are allowed in the location installed. The code is easy when it declares prohibited, it is a little more tricky to apply all the little restrictions, so I have to make a few assumption to show if you can reach a "yes" conclusion.

Romex is a brand, but the name generally refers to type NM cable, which is covered in NEC Article 334.

Flexible conduit usually refers to type FMC, covered by Article 348, Sectiion 348.10 says permitted in exposed and concealed locations, then 348.12 says when not allowed. Nothing implies wire or cable types. Continue to section 348.22 (or paragraph XXX.22 for respective conduit type) where it specifies number of conductors, second paragraph says "Cables shall be permitted where such use is not prohibited by the respective cable articles. The number of cables shall not exceed...Table 1, Chapter 9."

So you need to go to the respective cable article, section 334.12 Uses not permitted.

334.12(A)(1) say only allowed in..334.10(1)(2)(3), (5)..., those are building construction types (wood, steel, resi, comm). None of the others restrictions in 334.12 address use in conduit.

So you apply both sections it says you can only use a cable in conduit when both the conduit and the cable are approved and you don't exceed Table 1 Chapter 9, as long as you don't fit any other prohibition in XXX.12 for the cable or raceway you want to use. NEC Chapter 9 Table 1 Note 9 says that the size and fill calculation of ecliptical cables is calculated as a single round conductor using the major diameter. The calculation results in a larger diameter cable than you would expect.

334.15(B) only specifies minimum length when a sleeve is installed strictly for protection.

What happens sometimes people get confused, they pull into a buried conduit, and are told they can't do that. All underground is considered wet location, NM is not permitted in wet locations NEC334.12(B)(4). The interior

of the conduit doesn't qualify as dry location (300.5, 300.9). NM cable or the conduit, aren't really the problem, it's where you are installing the NM cable.

You can't remove the outer jacket of NM because you remove the ANSI/UL marking identifying the class listing of the wire. I've never seen that on the individual conductors. MC cable for instance has a tell-tale strip for that purpose.

Generally not, but it's allowed in some circumstances. For example, in exposed areas to protect the Romex from damage (such as a cable run from the ceiling to an outlet in an unfinished basement).

NO! It is NOT OK! Unless you remove the other insulation of the Romex where the wires will be in conduit. "Double shielding" the wires will make the wires get hotter. The Romex wiring is approved as it is sold for use inside walls, only. So remove the outer insulation and then you can run the wires through conduit. If your Romex is run into walls and then through a conduit, you leave the outer Romex jacket on the area of the cable that is in the wall!