Is there an attitude limit for firing a missile from a fighter jet? Can the pilots fire them when they're pulling Gs, or inverted?
The combat advantage of fighters (as opposed to heavier-payload bombers) is their maneuverability, therefore a missile delivery restriction such as g-limit (think Sidewinder heat-seeker in a dogfight) or attitude would unnecessarily restrict their efficacy.
As a Vietnam-era fighter and attack pilot I fired both unguided (aim and shoot rockets) and guided (target lock-on and shoot) missiles, and neither had associated attitude limits. Missiles that fall clear of the weapon station before rocket-motor ignition (such as the TV-guided Walleye or the optically-pilot-guided Bullpup) are obviously restricted from negative g to ensure safe separation from the airplane, i.e., negative g release from an under-wing weapon station would all but guarantee impact with the airplane.
Yes. Specific weapons can be employed within specific limits and load factors of the flight envelope, some of which includes inverted flight for air to air missiles, as this F-22 test pilot demonstrates during the Raptor's flight test.
Yes, Modern air to air missiles can be fired while manoeuvring but it can reduce kill probability. So missiles can be fired when aircraft is inverted but within certain g-limits. Firing a missile while manoeuvring also depends on altitude of aircraft, type of missile and effectiveness of seeker.