What is the ruling on contacting the Christian owner of the internet service to fix an outage, knowing that my siblings use it to download movies, while I use it for beneficial things? And what is the ruling on continuing to subscribe to the service if the owner intentionally cuts off the connection sometimes without deducting the cost, knowing that my unsubscribing might anger my father?

It is permissible to deal with non-Muslims in permissible transactions. Providing internet service falls under the category of ijarah (leasing/hiring), and it is a condition that the benefit be permissible. If there is an agreement to use the service for what is permissible, then the contract is valid, and the lessor is entitled to the rent. The rent should be reduced proportionally to the period the service was cut off, unless there is an agreement to the contrary. If the questioner does not provide the service (i.e., is not the provider), then he bears no sin for others using it in disobedience to Allah, for "no bearer of burdens will bear the burden of another." He should, however, advise them to fear Allah. His connection to the repairman is not considered a cause for their disobedience, because the disobedience stems from their actions, not his.