Light twin pilot reported the landing gear would not extend normally even after employing all available techniques. The nose gear collapsed on landing; resulting in damage to the aircraft.

Narrative

Prior to take-off; per the preflight checklist; we tested the hydraulics by pushing the gear lever to the full down position. After engine start; the gear lever moved into neutral position; indicating positive hydraulic pressure. Departed ZZZ to the practice area for maneuvers. Climbed to 5500 ft. While performing a drag demonstration; the flaps failed to extend. We suspected hydraulic pump failure and tested the landing gear. It failed to extend as well. We used the hydraulic backup hand pump with the gear lever in the down position per the POH. After pumping the hydraulic hand pump approximately 20 minutes (probably 800-1200 pumps); the two main gear indicated down and locked with two green lights. The hand pump lost the feeling of positive pressure and the nose gear light never indicated down. We attempted to lock the nose gear into place using G-force by doing steep turns; abrupt maneuvers and pulling positive G-load with no results. We kept trying the manual hydraulic pump but it still had no feeling of pressure. We reviewed the POH for all gear backup procedures. We climbed to 7000 ft; flew straight and level and activated the CO2 emergency gear deployment with gear lever still in the down position per the POH. CO2 caused the nose gear light to activate; but then the two main gear lights turned off. We could see that the nose gear was extended in the mirror. We called ZZZ Center; advised them of the situation and requested a flyby of the Tower for a visual inspection of the gear. The Tower reported that all three gear appeared to be down. We requested to climb over the field to attempt to extend the flaps. We attempted to extend the flaps with the hand pump per the POH; but since the CO2 cartridge had already relieved all hydraulic pressure; the hand pump would not work; even if it was functioning correctly. We planned a no-flap; soft field landing procedure and briefed what we would do if there was any indication of gear collapse during landing. We touched down briefly; felt a dip to the right and called go-around. After the go-around; the two main gear lights turned green and the nose gear light turned off. We assumed we had knocked the main gear into locked position with the touch and go. Since the nose gear was previously green; we did not think it could have unlocked itself; with the gear lever still in the down position the entire time. We thought there was a good chance all three gear should be locked in position at this time; even though the nose gear light had turned off. We could still see the nose gear was down in the mirror. The Tower still reported they could see all three gear appeared extended. We planned the same kind of landing again and briefed what we would do if the nose gear collapsed. We touched down softly on centerline; but as the nose descended; the props struck. We executed our assigned tasks; turned off all fuel; fuel pumps; master; mags; avionics and exited the plane as quickly as possible. There was some smoke in the cockpit and we saw fuel leaking under the wing as we exited. We ran off the runway around 300 meters away. We called Tower and reported the incident. Tower advised that they had already deployed the Airport Rescue and Fire Fighting (ARFF) response team. ARFF found no fire. The aircraft was subsequently towed back to the ramp. There were no injuries to report.