Poloniex has mobile Apps that you can download, maybe that would also help you monitor your order.
If I place a stop-limit-order and than log off and shut down the computer:
Will the order still be executed?
Sorry for this fucking dumb question

Others have already answered your questions, so I will offer you some advice, first, make sure to set up a decent password, activate 2FA, and NEVER leave significant amounts of Cryptos in exchanges for too long, of course, thats for the most paranoids, but if you don't control the private keys, then they aren't really your funds.
If someday Poloniex acts differently, we'll know from the posts of some users here to warn us all and hope that it won't be too late for us to withdraw the funds.