Putin vs. Navalny.

What do you think about the election in Moscow?

In your question, "do you think" is much better than "are you thinking".  Also, because you are surely talking about the specific mayoral election that recently took place, the definite article "the" is required.  And because Navalny only ran (баллотироваться) for one office (должность), "election" is better than "elections".

Putin vs Navalny.

And what are you thinking about elections in Moscow?

Good work, minor point, it is not formal to start a sentence with a conjuction like "and" or "but".  This is changing however. 

As to my commentary regarding your question.  I do not know anything about Navalny, but as far as Putin goes...he strikes me as an oligarch who has not stomach for political dissent.  I agree that liberal democracies are never pretty, the sausage making is front and center, and they fail regularly in small ways. However when an oligarchy fails, it is best to find the exits as soon as possible.