Indeed! Breathing together in the literal, but breath also carries connotations, as often expressed in the Greek pneuma - the Bible, according to Paul, was "theospneustos", God breathed, the soul is the "pneuma", so conspiring implies an intermingling.

Conspiracy in general, yes.

"Conspiracy theory" is another thing, more specific than conspiracy.

I think the implication is a bit more literal. When you conspire with someone you communicate secretly with them in order to plot and coordinate your actions. You do this by being in very close proximity to them and whispering, so close that you are sharing breath i.e. breathing together.

investors conspire in public to take over voting power, for instance, and do so in public to attract others