If I transfer BTC from an exchange to my wallet which is on an offline computer, what happens? I will bring it back up in a week but what happens to the BTC, do they not send or stay still? Thanks.

Has anyone used OKPay before? Getting verified by them seems like a lengthy process. What I'm interested in knowing is how trustworthy they are as a service and if they are better for transferring money to an exchange than say do a direct bank transfer.