We're not told he has any brain damage.
And furthermore, we're not told he does not have brain damage. All the talk is about his "mind", and we should agree by now that "mind" and "brain" are two different things.
...an inability to comprehend language
An inability to produce language would fit the bill here. Aphasia can come in all the obvious variants: "input failure", "output failure" and "input and output failure". Pike could have had the middle sort, understanding what was being said but being unable to respond in kind.
I never got the argument about the chair being "poor". It's a prop like any other TOS one: simplistic on the outside, but credited with great capabilities. Although just like you say, it's not explicitly credited with much in just one episode, any more than the tricorder or the phaser was; the full extent of the capabilities of these latter pieces of tech came only after an exposure of dozens of episodes.
The seeming lack of manipulative or obstacle-crossing abilities might have gone away had Pike continued as a recurring character, too. Much like the Daleks or R2-D2 gained those abilities after a few initial appearances where such shortcomings seemed overwhelming.
Anyway, a few hipshots at the details:
How does Pike eat?
Given how immobile his face appears to be, his mouth is probably incapable of handling food the usual way. So the two basic options are bypassing the mouth altogether (plenty of room in the chair for an IV feerding system or even an artificial stomach) or having a nurse squirt liquid food into the passive mouth.
How come he doesn't have a neural socket to connect his brain directly to a computer?
Again, for all we know, he does. And that's what allows him to beep yes or no.
That nothing external is shown is actually a good thing, just as with the neck brace: it's more plausible that way.
What exactly did they do to test him to be sure that his mind is coherent, that he can still understand language?
There are some real-world situations where such coherence is tested exactly like Pike's would - namely, quiz games where you are only allowed to answer "yes" or "no"!
Severely paralyzed people are also communicated with by observing primitive interaction means such as blinking. It would be impossibly complex to convert those means into, say, Morse code, but it's quite possible to assess the clarity of mind of the victim merely by using the interaction at a very low and simplistic level, such as "yes/no", "more/less", "right/wrong".
Timo Saloniemi