I will let others comment on grammar, however, as a speaker and reader of BrE for some reason starting the second sentence with, "It's" rather than, "It is", seems to jar with me when I read, or say it. It may of course be fine in AmE usage.
I think it is because you are so emphatic in your second sentence, that a government is vital to a country's civilization that, to me, it needs to start with the more emphatic "It is....":
"How important is a government to its country? It is the foundation of the country's society.......", to me using "It's" lessens the emphasis and so lessens the delivery of your point of view". (i.e. you put forward a strong argument using a weaker delivery, which to me sounds and feels odd).
A subtle difference I know, but to this BrE it makes a big difference when reading, or saying it.
This is my personal view, others may, no doubt, have other views, and as I've said it may be perfectly fine in AmE.