1. My brother came and pushed me away.
2. My brother came and pushed me off.
[Can we use both 'away' and 'off'? Are both sentences the same in meaning?]
3. What field of study are you interested in?
4. I'm interested in biology.
[Is this answer suitable?]
5. When we were waking around, a foreigner came up to me.
6. When we were waking around, a foreigner came to me.
[What is the difference between 'came to' and 'came up to'?]
1 and 2
"away" and "off" have different meanings, and "off" will usually need an object (since "off" is a preposition).
My brother pushed me away. <~~This is fine.
My brother pushed me off the deck. <~~includes use of an object
3 and 4 are fine, yes.
5 and 6
I think you mean "walking" in both sentences, right?
There's very little, if any, difference between "to" and "up to" in these sentences.