Electrical – enter the side of the panel *through* the stud

Note: All code references cited in this answer are from NFPA-70:2014 (the 2014 NEC).

Question #1: It looks like you have it right that the installation is in violation of 312.5(C). The code reference is attached at bottom of my answer.

• 312.5(C): There is less than 18" of EMT used. This rule was evidently introduced in the 2002 NEC because a number of electricians were stuffing a bunch of NM cables through a single hole. I have yet to exactly figure out from code history why 18" was the selected value, but I suspect it is because a generally accepted height that is no longer subject to physical damage is 8' and the maximum height of a breaker is 6'7" (hence the potential minimum height for the top of a service panel with a breaker at this maximum height) per 2014 NEC 404.8 (8' - 6'7" = 17"). It seems like the installer met the spirit of the rule given the low basement ceilings (but not the letter) by giving each cable assembly it's own EMT protection even though the length is only 8" -- this is perhaps why the inspector let it slide.
• 312.5(C)(a): At least one cable is not secured within 12" of exiting the EMT (the orange one on the right).
• 312.5(C)(d): The EMT is not sealed at the outer end (per your description)
• 312.5(C)(e): There is no visible NM sheathing extending into the box and is definitely less than the required 1/4" of sheathing extending into the panel

Question #2: I could see an argument for installing NM cable outside of conduit so long as the cable originates from a punch-out toward the middle of the panel. The reason I say this is that you effectively have EMT providing protection against physical damage on all sides around it. That call would ultimately be up to the inspector's opinion though -- the same goes for all calls on "subject to physical damage" since the NEC doesn't define it.

Question #3: You will note that flexible conduit is not expressly allowed by 312.5(C) and expressly specifies only two allowable methods: securing the cable and specific uses of nonflexible raceways. Also, all flexible conduit options are subject to the same "cannot be used where subject to physical damage" restrictions as NM, so it wouldn't do any good in this case even if it were allowed since the NM cable is being protected because it is subject to physical damage.

Just to show the specific prohibitions on the various flexible conduit options:

ARTICLE 348 Flexible Metal Conduit: Type FMC

348.12 Uses Not Permitted. FMC shall not be used in the following:

(7) Where subject to physical damage

ARTICLE 350 Liquidtight Flexible Metal Conduit: Type LFMC

350.12 Uses Not Permitted. LFMC shall not be used as follows:

(1) Where subject to physical damage

ARTICLE 356 Liquidtight Flexible Nonmetallic Conduit: Type LFNC

356.12 Uses Not Permitted. LFNC shall not be used as follows:

(1) Where subject to physical damage

ARTICLE 360 Flexible Metallic Tubing: Type FMT

360.12 Uses Not Permitted. FMT shall not be used as follows:

(5) Where subject to physical damage

ARTICLE 362 Electrical Nonmetallic Tubing: Type ENT

362.12 Uses Not Permitted. ENT shall not be used in the following:

(9) Where subject to physical damage

ARTICLE 320 ARMORED CABLE: TYPE AC

320.12 Uses Not Permitted. Type AC cable shall not be used as follows:

(1) Where subject to physical damage

ARTICLE 330 METAL-CLAD CABLE: TYPE MC

330.12 Uses Not Permitted. Type MC cable shall not be used under either of the following conditions:

(1) Where subject to physical damage

I think that has it covered.

312.5 Cabinets, Cutout Boxes, and Meter Socket Enclosures. Conductors entering enclosures within the scope of this article shall be protected from abrasion and shall comply with 312.5(A) through (C).

(C) Cables. Where cable is used, each cable shall be secured to the cabinet, cutout box, or meter socket enclosure. Exception: Cables with entirely nonmetallic sheaths shall be permitted to enter the top of a surface-mounted enclosure through one or more nonflexible raceways not less than 450 mm (18 in.) and not more than 3.0 m (10 ft) in length, provided all of the following conditions are met:

(a) Each cable is fastened within 300 mm (12 in.), measured along the sheath, of the outer end of the raceway

(b) The raceway extends directly above the enclosure and does not penetrate a structural ceiling.

(c) A fitting is provided on each end of the raceway to protect the cable(s) from abrasion and the fittings remain accessible after installation.

(d) The raceway is sealed or plugged at the outer end using approved means so as to prevent access to the enclosure through the raceway.

(e) The cable sheath is continuous through the raceway and extends into the enclosure beyond the fitting not less than 6 mm (1⁄4 in.).

(f) The raceway is fastened at its outer end and at other points in accordance with the applicable article.

(g) Where installed as conduit or tubing, the cable fill does not exceed the amount that would be permitted for complete conduit or tubing systems by Table 1 of Chapter 9 of this Code and all applicable notes thereto. Informational Note: See Table 1 in Chapter 9, including Note 9, for allowable cable fill in circular raceways. See 310.15(B)(3)(a) for required ampacity reductions for multiple cables installed in a common raceway.

I would consider moving the panel down and bringing the cable in one of the knockouts on the back of the panel. That way the part of the cable inside sheath is a "straight shot", and need not be bent. Then you can remove the sheath and bend each of the individual conductors as needed.