To dismantle something would be to take something apart carefully. I The context of an argument this would mean disprove by means of carefully and systemically considering all the evidence. (As mentioned it is not a very common expression to use in English) To demolish something doesn't indicate quite the same level of care as to dismantle, and perhaps indicated a level of forcefulness. If you were to demolish an argument, normally this would be done in a very forceful way, perhaps with just one very convincing line of reasoning. I hope that helps (Also as a note - if you say he dismantled the argument, this doesn't necessarily mean that the action was completed like it would with demolish)

Both make sense, but demolish is much more likely than dismantle. 'Destroy an argument' is also possible.

Dismantled his argument means that you went through his points one by one and successfully argued against them. Demolished his argument means that you argued against him so he had no way of supporting his argument. So they are similar but 'dismantled' suggests you did it in a one-by-one way. Like you might take a phone apart slowly rather than smashing it to pieces.

That is not a phrase that would be commonly used in english.

best if you provide the full sentence , to have more context. I'd say "destroy ..." instead but again, it depends ....