Ad aspera

The dragon's stat checks are tripled – is this essentially just saving throws against resistances, and maybe some other checks during peacetime, and that's it?

The same applies to attacking, with things like strength versus strength for paralysis/disarm/decapitation.

Do different one-handed weapon specializations stack for Vidur, and do identical ones?

Both weapons work with the default specialization and their own specific bonuses. That is, with an axe in the right hand and a sword in the left, both will receive bonuses from the ranks in one-handed weapons, plus the sword will receive bonuses from the sword branch, and the axe will receive bonuses from the axe branch.

Or do you also have to finish off allies to end the battle?

No, you don't.