I was a little surprised when you mentioned you have apps for BBC and CNN, and although I understand we need to get our news from somewhere, these news agencies are being used to spread dishonest propaganda, and journalists on CNN have actually called for violence on many occasions. These two particular news sources are not only being used to spread propaganda (internationally), but violence and division -- which I fear is infecting the church. Divide and conquer is the current agenda, and it seems disciples are being influenced by misinformation (as evidenced in things they post on social media).
I hope you will be able to accept my response, even if you end up disagreeing with me. It's worse than you thought (BBC, CNN). I also have Reuters and Al-Jazera! I receive regular updates from NASA, the Christian Chronicle (mainline Church of Christ), Voice of the Martyrs, and several other sources. None is totally balanced or accurate, and neither are my own views of the world. For me, I hope for an approximation.
As Paul stated, "we do not wage war as the world does" (2 Cor 10:4). We reject the values of the world (1 Cor 1:18-2:16; James 4:1-4; 1 John 2:15-17). You are right to be concerned that the world is affecting the church, for the evidence for this is strong.
In the same way, I visit many nations that promote violence. I purchase visas that, in some countries I am sure line the pockets of unscrupulous officials. I suppose I could boycott them, but then I couldn't go anywhere ;-). And I interact with pro-violence people of many stripes, without always directing them to Matt 5 and Romans 12 -- sometimes there are more substantial issues.
You and I are in the world, but not of it (John 17:16). I trust that mature Christians are able to hear the news without necessarily swallowing everything they hear. Of course you are right -- since most people don't read history books (most don't even know where the nations of the world are on a map, let alone recognize persons or patterns), nor usually do they visit the places referred to in the news, or interview witnesses -- people jump to conclusions, and often hold opinions completely out of proportion to their knowledge. But that's a lack of humility. If we're uninformed, we should be quiet. In politics and religion (those two realms especially -- don't you agree?) opinions tend to be most out of proportion with knowledge.
As you note, everyone gets the news from somewhere. In Luke 13 Jesus describes two events -- it is highly unlikely he was present at either of them. Perhaps he used his miraculous powers to investigate, but more likely, as incarnate and human, he heard about this from friends or family.
We can get the broad outlines -- the big picture -- without necessarily being sucked in to the messages of the evil one. I firmly believe there was a school shooting in Broward County, Florida, a few months back. I remember some of the details, but none of them is especially salient. Even if there was a slant to the reportage, the actuality of the event itself would have been difficult to misreport.
I do not discuss politics in social media (FaceBook, Twitter, Linked-In, etc), and even at my website discuss issues only when there is a biblical purpose to do so. And I actively discourage Christians from getting into divisive social and political issues online. Or even face-to-face, unless there's an immediate threat.
Now, if there is something specific I have been teaching that you think has been influenced unbiblically by my efforts to keep up with the news, please let me know. It would help if you identified the doctrine. Otherwise, I believe you may be reacting. I would be grieved to think I was doing harm, so feel free to tell me if I have missed something. Otherwise, I'll keep on reading and observing. I think they help me better connect with people. They strengthen my convictions about the deceit and manipulations of power mongers. They confirm my faith in the Bible, too -- whenever I take a glimpse, things seem to be just as bad as Scripture says they are.