It sounds like you're talking about direct and indirect reported speech, which are something you heard someone else say (and similar sentences). In this case, all of your sentences are fine -- yes, native speakers often backshift reported speech they heard in the past.
All of these can mean exactly the same thing:
He told me he is going to the game tonight.
He told me he was going to the game tonight.
He told me, "I am going to the game tonight."
The only odd thing about the first sentence is the mixed tense, as you use the present tense for the first half of the sentence ("he has been receiving ...") and the past tense for the second half ("he was dealing ...") This is a common error with spoken English, if you are not thinking carefully about how to express a situation and you backshift part of the sentence but not the other part. Native speakers will understand what you are trying to say and probably not even notice.
However, if you value precise language, I suggest:
He did say he has been receiving your emails and that he is dealing with the matter.
Alternately, the deliberate use of the past continuous might mean he was dealing with the matter, before something else happened:
He did say he is fixing the car -- he was working on the radiator but then he noticed that the coolant was leaking from somewhere else, so now he is trying to locate the leak.
Anyway. To answer your second question: "the matter" is ambiguous, but the natural assumption is that it's some matter explained in the emails.
