Did he possess one of the Palantíri at Barad-dûr? Or could he just "tap into" Palantír conversations at his will?
Yes, Sauron possessed one of the Palantíri. There was originally a stone placed by the Dunedain at Minas Ithil, which Sauron captured and turned into Minas Morgul (the big fortress the Nazgûl that Frodo &c go past in their way to Mordor). Sauron would have taken that stone along with the fortress.
However, Sauron did have more "control" over the stone than others; the stones were not just mechanical devices. They were "aware" of their user, and in particular, that person's authority and ability to use the stone. Sauron was apparently "better" at it than Saruman or Denethor, which allowed him to exert some influence over them and what they saw in the stones. That's what eventually drove Denethor insane, and turned Saruman to Sauron's side.
Aragorn, on the other hand, was the rightful heir to the stone at Minas Tirith, so he was able to influence Sauron into coming out to meet him.
Yes he had a Palantir. You had to have another Palantir to communicate with someone else using one. There was only one Stone, which was lost during the Kin-strife, that could eavesdrop on two other Stones in communication. Otherwise with, or without a Stone, he could not eavesdrop on two Stones in communication.
Denethor, like Aragorn, was a rightful user of the Stone. Sauron never, ever dominated Denethor through the Stone as he had with Saruman.
Sauron failed to dominate him and could only influence him by deceits. [The Palantiri]