This passage actually refers to bonding. The reference is Article 100, which provides the definitions of grounding and bonding. Grounding in this passage makes no sense from a technical standpoint, so the correct word is bonding.
 

[250.114]. Again, we have a word misuse. The equipment must be connected to an "equipment grounding conductor" which is, in actuality, an equipment bonding conductor. There are a few exceptions to this requirement, but generally it applies.
 

[250.118]. No. It doesn't even have to be wire. See the 14 types of EGC in 250.118.
 

[250.122(A)]. Yes. It can't be smaller than shown in Table 250.122, but doesn't need to be larger than the circuit conductors supplying the equipment.
 

[250.122(F)]. Yes, if you run the ungrounded conductors in parallel.