Responding to "Petition…"
I did not think Bastiat meant that imminent death would happen to all of mankind if they found a way to extinguish the sun. I understood him as implying how even something so illogical could get government and law enforcement overreaching over something that is inherently "natural" to the world and already provided freely. This predisposes us to the contradiction that is the government (from his point of view) before we read about it in Government and Law. This also makes your analogy a bit skewed for me. Although you provided a new lens for me to see "The Petition"
"The issue is that if a cure for cancer is revealed and made accessible, there will be a massive economic loss for doctors, hospitals, and especially oncology specific practices."
This part got me to analyze the reading from an economic market perspective. In particular, how sometimes the systems own needs (in economic market sense profit) can go against the very interests that spanned the system to begin with (capitalism for consumer economic power and mobility due to a "free-er" market). This is a nice economic interpretation of the contradictions that Bastiat laments on through "Law" and "Government".
Responding to "Government"
I don't think Bastiat is particularly contradictory; I would argue that the majority of his writing is him parlaying the contradictions he's observed to us in a self-insertable writing style that can appear contradictory. From the beginning of government, Bastiat echoes all of the speakers' and press's cries, with half of them contradicting themselves within their very own demands.
Responding to "Law"
I fundamentally agree with your thoughts and opinions on "Law." I found your use of current day law enforcement to explain your stance and thoughts effective. It was a good way to say what Bastiat was stating, while explaining why your stance was what it was.
"These people are permitted to carry guns and even fire them without facing punishment if the situation permits this. I would not want to live in a world where any random individual can act in the way that these officers act."
Those were really strong realities that helped contrast the differences between the demands a government must meet and how internally contradictory they could be. Which you go on to acknowledge this dilemma, even admitting you do not know the solution [I do not know who would]. It was great how you further showed the societal implications that can permeate the government and law enforcement choices by mentioning the unjust actions of police officers.
I found this a great read! I love the amount of opinion throughout and your willingness to have discourse!