"My first question is: does the code allow NM cables connected from the service panel in this manner?"

The NEC requires a cable clamp were NM exits the box.

If the conduit is not part of a complete conduit system but is only for mechanical protection you do not need a junction box since you are not changing wiring methods (conduit to NM).

If the conduit is only for mechanical protection the normal conduit fill rules also do not apply.

Look again and see if you can exit the back of the panel with the new cables and run them entirely inside the house
It would be a far cleaner instillation.

Thanks for the response brickeyee - I understand your comments, but there are still a couple of thing I am not sure of. First, I know that cable clamps are required when NM exits a j-box etc, but in my case the cables are exiting the back of the service panel and then through holes bored in the framing structure. I can't see any way to clamp these cables in the panel. The hole at the back of the panel is 2" diameter, there are 4 cables, and I don't understand what kind of clamp I could use. If this configuration is 'legal' (that is, no clamps are required) then I certainly will be able run the rest of my NM cable through this hole in the same manner (as per your suggestion) - but the issue of clamping is what I don't understand.

Those cables should not have been entered into the panel in that manner.
You need to go to the electrical supply house and get a 3) romex connectors for the 12/2 mn cable, knock out (3) 1/2" holes and re-enter the 12/2 cables into each hole.
You will also need a 3/4" romex connector for the 6/3 mn cable, now knock out a 3/4" hole in the panel and do the same as the other cables.

Thanks christophersprks - that is exactly what I was afraid of. Just to make sure I understand the rules: a) the only way I can connect NM in the service panel is with a cable clamp for each individual cable, and each cable must be in its own knockout. b) if I use condiut with the service panel I can NOT use cable, only single conductor THHN (because of the clamping requirement with cable)

A cut out box (the large box used for the panel) is the same as any other box.

NM requires a cable clamp when entering any enclosure.

Even the plastic boxes use the small tab in the opening to grab the jacket of the NM and fulfill the requirement.

There are cable clamps rated for more than a single cable, so the "individual knockout" is not correct.

You can use conduit with cable for mechanical protection.
You can affix a clamp to the end of the counduit to secure the cable.

I've seen panel installations where most or all of the NM exits the panel through a 2" or larger conduit. There is no clamp on the conduit but the cables are stapled near the exit point of the conduit.

I couldn't find explicit support for this in the code. But I'm no code expert, so I asked about this method on this forum a couple of years ago, and was told then that it was compliant. Also, I recall that at least one of these had an "approved" sticker on it, so it was acceptable to an AHJ here.

What Ecranny describes sounds like it could be a horizontal application of this same idea - IF the cables are in a conduit, and IF the cables are properly secured inside the wall where they exit the conduit.

Or am I wrong about that?

"I've seen panel installations where most or all of the NM exits the panel through a 2" or larger conduit. There is no clamp on the conduit but the cables are stapled near the exit point of the conduit. "

You are unlikely to find this in the NEC since it is actually not in compliance.

It is a bad shortcut used to save the time of running the cables through the correct clamps.

Even if a cable clamp is present the cable must be fastened near the box exit unless it is fished in place.

I've seen panel installations where most or all of the NM exits the panel through a 2" or larger conduit. There is no clamp on the conduit but the cables are stapled near the exit point of the conduit.

See the exception to NEC312.5 (C). This allows you to use feed NM cable through conduit in to an enclosure. The conduit must enter through the top and be between 18in and 10ft in length. Other specific rules apply, but it can be done.

This of course does not apply to the OP's situation, but I have yet to run across an inspector that would fail that type of installation as long as the cables are stapled at the first reasonable opportunity after exiting the conduit.