From what I've witnessed:
Pirates will raid exactly one player every time the counter runs out, even if there are multiple bases (in multiple systems even).
Prates will raid the player with the highest bounty on them.
Pirate bases are rendered nonfunctional when the planet is bombarded to the point the pirates loose control of it (or if the loyalist advent titan captures it), this means that the TEC superweapon can take out pirates fairly easily. The ships and defensive structures are irrelevant, except that they make it harder to conventionally bombard the planet. The pirate base can not be de-capped via culture.
When a pirate raid is finished (system is wiped out) surviving ships will return to the pirate base they launched from and become additional defenders. This includes pirate raids launched via buying missions from the pirate faction.
Buying a pirate mission sends a raid of strength depending on amount spent at a particular player.
TEC rebels have research that allow them to become allies with neutral factions and pirates. Once this tech is researched the TEC rebels can no longer be valid pirate targets and can pass through pirate bases.
Here's the things I'm not sure of:
Pirate raid strength increases as the game progresses. I am unsure if this is based on time or bounty collected by pirates.
The strength of individual pirate ships and structures increases as the game progresses. My assumption is that this is based off of bounty collected by the pirate faction, and it appears to apply to all pirate bases.