What are my options for getting this mess of cables under control?
I'm much more concerned about the fact that you have split-bus (Rule of Six) panel.  I guess I wouldn't worry if breaker sum &lt;= service rating or at least within 20% of it... the reason we dislike split-bus panels is it's possible to laughably overload a service.   You only have 4 of the 6 breakers in-use, so that helps.
I'm also concerned with breaker spaces, I don't see how to power a house with so few breakers.
Other than that, I love CH panels. Great panels.
A few defects:

The 12/3 cable with shared neutral needs to go to a 2-pole breaker.  This is a Multi-Wire Branch Circuit and must follow those rules

The 10 AWG circuit needs to be breakered 30A not 40A.

The 12 AWG circuit needs to be breakered 20A not 30A.


Tell us more about the last two; rarely certain motor loads qualify for a breaker bump not otherwise allowed.

By code, am I allowed to route 5 NM cables together and run them through a single 2” knock out?

Yes, if you have a proper cable clamp rated for 5 NM cables.  NEC 110.3(B).
That might be a unicorn.

Am I allowed to run 6 circuits in a single conduit (as long as the conduit is large enough) or is there a maximum number of circuits?

Only if the a) length is less than 2 feet...  or b) the conduit is &gt;4 square inch cross section and &lt;20% full.
Otherwise, since you have more than 4 circuits in the conduit,  you must derate #12 wire to 15A, #10 wire to 20A, and #8 wire to 28.5A.
There's a lot of complicated math behind it, but that's the gist. See 310.15(B)(3)(a) and how it interacts with Article 334's derates and the statutory minimums in 240.4(D) and round up/down in 240.4(B).

Am I allowed to connect conduit to the bottom of the electrical panel or do I have to connect to the top?

If the factory provides knockouts, you're allowed to use them.  Other holes, see 110.3(B) &quot;must follow instructions and labeling&quot;.
That clause (with 110.2) invokes the UL White Book and its design rules.

If I use PVC conduit, do I need a separate ground wire for each circuit or can I combine to reduce the total number of wires?

Everything you have talked about is NM cable.  You need 1 ground per NM cable.
If you were using wiring methods that entail individual wires, then grounds can be shared.

If I use a metal conduit, do i need to include a ground wire?

Nope, the metal conduit works except for certain flexible types. Assuming, of course, you are working in individual wires such that you could share a ground.

I would prefer if the new junction box was flush-mounted, but I assume it will need to be surface mounted like the electrical panel. Is there any way to connect conduit to the face of a junction box, rather than to the side?

No, because you have to be able to remove the cover without disassembling any part of the building 314.29.
You could flush-mount and then use a box extension to simulate a surface mount, for attaching the first conduit.
What's your fondness for surface mount?  That's been bugging me honestly. My hunch is that your biggest concern is aesthetics.  So why go to all this trouble? Why not just build a false wall in front of all those cables which you find so ugly?  Rip some 2x4's to give you the correct spacing at each location, then lay some drywall across it and Bob's your uncle.  You're allowed to hide cables. You could bring the false wall up so it's near flush with the panel.

By code, how large does my junction box need to be?

NEC 314.16 I believe. You need 2 cubic inches per #14 wire, 2.25 per #12 wire, and up to 5 per #6 wire. Grounds count as 1/4 of a wire (rounding up).