In the previous sentence the writer is talking about his Chinese lessons in the classroom, and how they were slow and laborious. Mr Li has not previously been mentioned, but do you think that it's maybe implied that Mr Li is also the classroom teacher?
君 is used as a title for "colleague", thus, 李君 is not a teacher but a fellow college student (probably form China).
Afterthought: Or perhaps 今度は is just contrasting the lunchtime conversation with the lessons
Yeah, that seems correct.
Could you explain the purpose of に here in そこで彼が言うには?
~が言うには is, I suppose, a kind of set phrase meaning "what ~ said is...". It's an expression preceding a quotation. Interestingly, any other verbs besides 言う, 語る, 話す or like that don't have this meaning, i.e., 彼がするには doesn't mean "what he did is...". I'm not sure but it would be from classical Japanese.
If 二、三人 (or "we") is the subject, then could you explain the use of で in 二、三人で励まし合いながら... Why not が?
The key is the compound verb ~合う. …で~合う means "to do ~ (to) each other among...".
e.g.
友達同士で助け合った
We helped each other among friends.(We = friends)
味方同士で撃ち合ってしまった
We/they shot to each other among the same side.(We/they = people in the same side)
Or, we can simply think it as the subject omission, since the subject is obvious from "…で". (が is acceptable, but sounds as a more objective expression than で.)
私にも日本語を勧めて = (he) recommended Japanese to me also (i.e. recommended that I also study it)
きた = ??
In addition to this, ~てくる expresses an action/movement is done towards the speaker/writer, not only a physical action/movement. Thus, it emphasizes that he recommended it
to me.