저랑 같이 배드민턴 치실래요? - to an elder. 우리 배드민턴 칠래? - to the same age or younger. '치다' means playing in the sentence.

'한테' is similar to the expression, '에게'. It makes the word been adverbial in Korean grammar. (I'm sorry, I'm not good at English grammar, so you couldn't understand what I'm saying..T-T)

for = ~를 위해(서) The meaning is right but it's not usually spoken to the letter. 1) 나는 나의 강아지를 위해(for) 음식을 샀어. (This sentence is a little unusual..) → 나는 우리 강아지 주려고 음식을 샀어.(Free translation is more natural.) 2) 나는 프리젠테이션에서(for) 사용할 노래를 발견했어. 3) 너는 내가 그를 위해(for) 그것을 하길 원하니?

It means, welcome to your visit.

There's no 천천하다. So you should exchange the word, 천천히, instead of 천천하다. 천천히 means slowly and it is adverb but 느리다 is the adjective. 천천히 and 느리다 have no diffrences in meaning. Just 느리다 has a little negative meaning.

"You leave first." could be an inappropriate expression if you are with an elder. Just use it with your friends. :-)

Is your teacher Korean? Could she understand the text totally if the contents are written in Korean?