1) "Teaches" is pronounced "tiːtʃIz". Two syllables, right?
....It's /ˈti:ʧəz/ in BrE; there are two syllables.
3) Supposing I am having my hair cut when my friend phones me and asks where I am. I reply "I am at the barber's." OR "I am in the barber's."
wWhich one is correct or suitable?
.....Both are possible.
4) If you pot this ball, I will concede. Correct?
.....Yes
6) You need not worry about my studies. Is it correct [...]?
.....Yes
OR we should use "to" as
in "You need not to worry about my studies."
.....No