1.) Yes. They are behaving like pirates but they are no longer officially pirates - hence why other pirates refer to them as government dogs who’ve given up piracy.
"Official pirate" is a concept you just made up. They are explicitly described as pirates in their current condition and their deal with the government consist on benefiting the institution with their piracy. Even then, an unofficial pirate would still be a pirate.
2.) no they weren’t they’re introduced as schibukai
They are both introduced as something like "A member of the Ouka Shichibukai. THE PIRATE Crocodile/Trafalgar Law".
3.) that’s an moniker - why would the government not want a famous pirate and remove their renowned name - doesn’t make sense.
Hancock has other renowned names that don't explicit her condition as a pirate. The reason why she's still addressed as Pirate Empress is because she's an active pirate.
4.) provide a scan please
I won't look it up for you. It's the panel in which Crocodile assures that he's a completely different kind of pirate than Luffy because he's down-to-earth instead of a dreamer.
5.) Other pirates refer to them as Government Dogs - but I’m sure there are times they are referred to as a pirate
A government dog can be a pirate too. The very reason why they are addressed as dogs instead of employees is because they are pirates who don't respect the ideals followed by the rest of pirates and try to chase them, therefore "traitors" within piracy.
So I'll repeat:
-System described as made up by pirates who perform sanctioned piracy, but piracy still.
-Addressed themselves as pirates.
-Introduced as pirates in official boxes.
-Use epithets describing a pirate condition.
-Addressed as pirates by tons of characters either directly (Smoker, Cavendish, Fujitora, Kizaru, Doflamingo, Sengoku, Rocinante, Kuzan, Tashigi, Brannew...) or indirectly (such as Smoker saying that Dressrosa became an island of pirates because of Doflamingo's reign, Buggy being the captain of a company of pirate mercenaries, Bellamy saying he wants to be a member of Doffy's pirate crew or Monet claiming that his captain will raise to Pirate King).
Should I go on? How come they aren't pirates?