Yes you should take your shoes off going into the Blue Mosque, because it is simply a mosque and not a national museum, like Hagia Sophia. You can find out all about it here:
I entered that mosque with my American friend, we played with the cats in the entrance, then there are plastic bags that you can take over there, you put your shoes in the bag and get inside, he was an american but I am Turkish, they lend you some headscarves but I didnt take them and got inside, it was no problem but would be nicer if you go inside with a scarf, why dont you buy one, there are many stores around and not expensive, would be a nice travel gift for yourself, I wasnt shunned or anything but please do not enter with short pants or bare shoulders, it is not a museum as someone said, it is a mosque and when you enter there you will see people praying
Yes, you should take off your shoes just like in any other mosque all around the world. And yes, worshipping places are supposed to be visited with no payment.
Yes. Mosque is a holy place for Muslims.
You are required to remove your shoes and women should be 'decently' dressed, covered arms and no mini skirts or shorts! Women are also required to cover their heads. Before anyone gets on their high horse, most RC churches who have a lot of non-Catholic visitors (in Rome, for instance) have a similar dress code for women.
Yes, you do. It's a place of worship.
And before you start griping about what killjoys Muslims are, I'd like to point out that in Italy, you are required to have a dress code when you enter a church, and in countries where Orthodox Christianity is practised, women are also required to cover their heads in churches.
Google it you moron