Hola. If you don't mind I am a man and native, I'd like to give you some information in reply to some of your questions.
First of all, it is impossible to find a safe and liveable accomodation for 20$. There are such cheap places, but you may receive insistant male guests knocking (!) on your door and i don't tell the atmosphere of the neighbourhood. Sorry but Turkey is not cheap any longer. I suggest to make a few good friends for sharing a flat.
Sorry I have no idea about ticket prices but I'm sure somebody here will tell you.
Don't worry about tattoos, nowadays it's very trendy in Turkey. My wife and I have tattoos, and one of my best friend is a tattoo artist. I'm helping him preparing the English version of his web site and running the communication with international customers.
Ankara is the capital and second largest city of the country; full of universities which means full of young students. Also don't worry about what to wear, you can wear anything as you like unless when hanging around the outskirts of the town.
Ankara is very cold in winters; snowy a few days but prepare yourself for (-) degrees.
In spite of the fact that islamic style of living is trendy nowadays, still your life will not differ too much comparing to Mexico. Smoking and drinking are the integral parts of a youngster's daily life
, but in the right time, right place. Don't worry, you'll learn everyhting so quickly.
Please forgive me but the last question is the funniest
What do you think for god's sake, having a journey to the past centuries? Many brands and models available, I recommend Orkid as brand, chose the model according to your personal criteria
I'm sure you'll enjoy your time
Come whoever you are...
just come as you are