I really, really wouldn't.
Absolutely, it would be better to leave your coins on exchange rather than entrust or send them to an unknown website.
Which website/place is trustworthy enough? Any suggestion?
Exchanges will not hold them indefinetelly. At least Cryptopia show that all coin must be withdrawn till 22/5/17.
None, but you can withdraw your coins on old hyper wallet address in your desktop which you hold the private keys. Then import those private keys into the new hyper wallet once it is finished. This is safer than sending your coins on a certain website.