Re: reported speech
Yes and yes, but...
Changing "this time" to "that time" is something I should have done the first time. Good catch.
The double "that that" isn't really a problem for me - they are two different kinds of "that."
The substitution of "now" is harder - actually "at that time" would be the most logical thing to say, but then you have an incredibly awkward double use of "that time" with the first time meaning "moment in time" and the second time meaning "instance" or "occurrence." So you may have to substitute something with the same meaning, which is allowed in reported speech if it keeps the same meaning.
He said that I could go home at that time, and that, in this instance, he would remain.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.