Демшиза = democratic + schizophrenic
Демшизоид = a person belonging to "демшиза".
It's a slangy word using when you need to describe the point of a person who sequentially disregards common sense and logic for the sake of "democratic freedoms", "democratic values", "political correctness" and so on. One may consider this behavior as a mental problem.
So, "демшиза" is 1) a group of radical fanatics (a bit anarchistic and "religious looking" liberals), 2) a special "state of mind" or "ideology" or 3) your judgement (as a characteristic) about some political facts, opinions, statements etc.
Answering your question,
Since this is a slangy word, you hardly hear it from officials and politicians. But some experts may use it as a "spicy" word. Ordinary people use it a lot, especially in the internet.
As to Navalny/Udaltsov, I can't say they belong to "демшиза", but we may say (if we see it) "это демшиза" in relation to their actions, opinions, statements etc ( meaning number 3, see above).