How to understand the sentence, "There’s no news like bad news"? and how to determine the meaning of "like"? Thank you for help!

What does "Better at teaching them what" mean? Thomas Jefferson and the other early American crusaders for public education believed the schools would help sustain democracy by bringing everyone together to share values and learn a common history. Home schooling forsakes all that by defining education not as the pursuit of an entire community but as the work of one family and its chosen circle, which can be great. Despite some drawbacks, there are signs that home-schooling parents are doing a better job than public schools at teaching their kids. But as the number of kids learning at home grows, we should pause to wonder： Better at teaching them what? Home schooling may turn out better students, but does it create better citizens? I think that it means "which way is better at teaching the kids". Am I right?

Is it a possessive case or a prepositional phrase? Allegations of corruption are a fact of public life in post-Soviet independent states. Is "of corruption" a possessive case or a prepositional phrase? I think it's a prepositional phrase, but not sure.

Am I right to correct the sentence by this way? Every spare moment of our time is being filled with some form of commercial connection, making time itself the most scarce of all resources. The corrected one: Every spare moment of our time is being filled with some form of commercial connection, which makes time itself the most scarce of all resources.