2/3. Apologies if I have asked before about these せ～ words, but could you just remind me what sort of language this is (e.g. old-fashioned, rural dialect or whatever)?
せねば is classical Japanese (未然形/the -nai stem of す + -ba conditional of ぬ for negative), and せにゃ is a dialect (derived from classical Japanese that remains in some regions). Thus, 話をせにゃならんな sounds "unpolished" as Hagrid's words, whereas 話をせねばならんな/ならぬな sounds "solemn" as Dumbledore speaks.
It's also the same as #3.
OK, thanks, by the way, I think I always assumed that ～ねば was a contraction of ～なければ (so that the existence of せねば would imply the existence of せなければ). Is this not true at all, or are these せ～ words an exception to the usual rule?
The negative auxiliary verb ない is a relatively new word. It started being widely used in the late Edo period. There are some hypotheses about the etymology (e.g. a Kantō dialect, derived from the negative adjective なし, derived from ぬ), but anyway ねば(or ぬ) is older than なければ(or ない), and なければ(or ない) is never attached to せ.