Verb +といて is verb(し)ておいて
that means (verb) して おく
and て is conjunction meaning "then"
This is a kind of "have done" form, I think.
I just remind you that I'm not an expart but ordinary citizen of Japan.
Hope you will soon figure it out :)
The とい, or toite form, is used for asking permission I think. I don't recognize all of the Kanji. It looks like more of a "tell me more" kind of question.
Read up on te form verbs, and conjugating verbs in Japanese.