{I just shuffled wiki's labels around and claim no special knowledge of Turkey. I'd guess Kemalists, despite their nationalism, would generally identify more with secular and democratic protestors than with the Islamist government}
Turkey has long (before Erdogan) hewed towards secularism in public life despite being a majority muslim country, generally guaranteed by the force of the Turkish army. Kind of like how Americans hope that if the US government ever turns on its citizens (or more than it has already), our brothers and sisters in the military won't turn their guns on us and will disobey orders.
Yes, but it's actually quite common in many countries. China has had something similar since the Communist revolution (though it has significantly died down over the years). North Korea still has a very strong "revolutionary tradition" in its propaganda.