Is my laughing at the Sheikh's mention of the state's established system — which the Sheikh mentioned sarcastically in the context of his discussion about the pillars of Islam — considered mockery of the religion?

It appears the questioner is suffering from misgivings (waswasa) concerning the issue of mockery (istihza'). He should cast away these thoughts and not indulge them, nor give Satan any power over him. Misgivings are a severe illness, and indulging them leads one into confusion, doubt, and distress. Just as one must fear falling into disbelief (kufr), one should also not be so afflicted by misgivings as to accuse oneself or others of disbelief.