Hello, Claire!

1. "bine" could be translated as "good", but words such as "okay","fine" can take it place just as well: este bine= it's good(it's okay); sunt bine=I'm good(I'm fine). The expression "Bine ați venit!" it's the equivalent of "Welcome!"(when you greet someone that's entering your house, for example). Therefore, if you find "bine" in an expression, it doesn't necessarily translate word by word, but as a whole.

2.I would translate "Ce faci?" as "How are you?" and "Ce mai faci?" as "How have you been?(lately)". The first one can be answered with "Sunt bine.", while the second puts an accent on what you've been doing lately, so an answer would be "Sunt bine, abia ce am plecat de la sală(I'm fine, I've just left gym). However, an answer with detailes for the second is not a must, so if you answer both questions with "Sunt bine." only, I think is just as fine. It depends on you.

3.Mă numesc(i think this one is used the most) it's like saying "Mă numesc Claire(I'm Claire)" but you can also use "Numele meu este Claire(My name is Claire)" that is equally correct. If you want to ask someone, you say "Cum te cheamă?", "Cum te numești?","Care este numele tău?". You can use any of them, but the last two I think show a higher degree of politeness. 

I hope this helps, and have a nice day!

salut Claire, I hope this could help you...

1. ce maifaci? ce faci? BINE = ok/ I am ok SAU well/ well done... colocation: Imi pare bine = I'm happy

2. subtle difference: ce mai faci? = how are you doing?/ how's it going? AND Ce faci? mostly What are you doing now? BUT you can understand "ce mai faci?" too, so it depends on the region...AND WHEN Did you see/have contact with the same person... because if you met him up regularly or yesterday, no matter the question you use, this person will not give you the news about him/her

3. Numele meu este Ion./Ma cheama Ion. Cum te cheama?/ Care e numele tau?

Spor la treaba/ good luck

PS. N-am diacriticele, I hope you will understand... if not, let me know