1. A health report is what it is. A certificate stating you are cleared all round in health from physical and lab tests done, etc. I cannot say what a fitness certificate is, please find out from your hospital if they are different but I do not suppose so.
2.I did not certify my certificate because I had no time to do it and there was no issue here about it. Things maybe different, and I do not want to advice anyone wrongly. I really cannot know these things so please call the embassy to find out.
3. Except they specifically asked you to, I do not think you are supposed to translate any document to turkish. I did not.
4. Aren't they supposed to book a flight for you guys? But if to answer, a round trip (to and fro) is usually more advisable but in your situation, I doubt you would know when next you may be returning(except you have an exact date u have planned for) so it is best you go for a one way. As far as I know, they are to fund your ticket somehow (I dont know if it is before you come or after), and it is a one way ticket to come and after your entire program here, a one way ticket to go back.
5.Coming to all that stuff, you are asking the wrong guy. I dont enjoy stress so i dont carry anything. I eat whatever i can find here. Unfortunately it is difficult to find Nigerian food stuff here like even yam. Rice is here ofcourse. Do not forget you have baggage limits at the airport and carrying these things maybe to your detriment. However, if you feel it is a must to carry certain food stuff to help yourself (remembering the fact also you may be staying in a dormitory where you are NEVER allowed to cook in anyway), and it will not outweigh your baggage limit at the airport, then by all means, help yourself. I can only advise you to carry along stuff like powdered milk(where it is hardly used in Europe), maybe grounded pepper, some noodles. Bear in mind these things would eventually finish and then what next?
CAVEAT EMPTOR
Guys, I cannot answer some of these specific official procedural/documentation questions the way you may expect me to because processes maybe different this year and I do not work for the Turkish Government. YOU WILL NEED TO relate with the person at the Embassy in charge of your coming here and ask these questions. However, personal questions like lifestyle issues here in Turkey ofcourse, you can go with my advice. I do not want to misguide anyone as I have always stated but I can only answer you based on my own experience 2 years ago. Stricter measures could have been enforced now in your time or less stringent. I cannot know. You may hear my own answer but please always call the official body to confirm and then make your decision. I am not responsible for anyone who does not follow official advice.
Cheers.