You need a blue pet passport (or whatever colour yours is) with a rabies jab.
Rabies is the only compulsory jab.
There is no quarantine - that I know of.
We use the land border with Spain regularly and take our dog backwards and forwards. There are no issues.
Although you are rarely asked for the passport, if you cross between Spain and Gib, you must carry it, or chances are you would be asked for the passport on the one occasion you didnt have it and your dog/other animal would be stranded
Customs are the ones to double check with.
If you move to Gibraltar full-time with a dog then it needs to be registered here. Ours is registered with the Junta de Andalucia (and is chipped too) as we origally got him at our place in Spain which we still have.
There are - sadly - no dog parks. Taking your dog out largely means taking him around the city streets, so please clean up afterwards.
Dogs are not allowed in the Alameda Gardens or on the Upper Rock nature reserve.
Hope that is helpful info.