I agree that time travel can make you go crazy if you think about it too long.  My biggest problem is the effect of the changes and the immunity of the to the traveler to those effects.  I can't imagine that traveling through hundreds or thousands of years of time could have no effects on the traveler itself, regardless of the effects to the timeline.

Ray Bradbury's "A Sound of Thunder" has as its theme this very paradox of time travel, as you call it. In this story, a man travels back to prehistoric times in order to kill a tyrannosaurus-rex that did die in the past.  However, in order to not disrupt anything else and reap different effects for the future, he must remain on a virtual path created for this purpose. Accidentally, he steps off and a dead butterfly returns to the present on the sole of his shoe. When the man rearrives in the present several circumstances are different as a result of his accident.

Stephen Hawking says that one of the laws of the universe is that cause must preceed effect. If you got into a time machine and killed your ancestor, the cause of your ancestors death (your killing him) would have come after his actual death. Hawking postulates that there's some kind of build-in mechanism within the universe to preven this from happening. 

This paradox is generally accepted. I don't really understand all of the physics and arguments behind it. Sometimes the argument is shortened to just include the first part. This means that we could theoretically travel back in time but if we change the past we change the future we came from. Of course, most theorist would argue that by going back at all we change the past so there's no real way around the paradox. Science fiction tends to ignore large parts of this paradox. It is a complicated theory. 

When the universe was planned and created, time was created along with it to give it a coherent structure. Time enables human beings to live in an unfolding universe in which they can develop and grow and change. This design would probably include safeguards that would make it impossible manipulate time and thus create the problems and paradoxes that you described.

I realize that some would not share my interpretation of this, but I believe the universe was designed by someone much more intelligent than any quantum physicist.

Al I can think about is Back To The Future. Doc repeatedly reminds Marty about the consequences of coming into contact with his parents and changing the future. While this is fictional, I cannot help but think that there could be some truth to it (if one could time travel, then they could alter how things would/could play out later).

Regardless, I see no way where true time travel could be made possible. Therefore, all we have to go off of are the assumptions made by the movies and interpretations regarding the problems which could be associated with time travel.

It all seems pretty implausible to me, but some pretty big names in physics, especially quantum physics, have tried to wrap their brains around it. My understanding is that this paradox is usally cited as a reason why time travel is actually impossible, but that some have argued for loopholes in quantum mechanics (which I absolutely do not understand) that would make it mathematically possible to travel back in time without altering or destroying one's self.

My opinion is that this is all just fantasy.  We can't travel in time so these sorts of issues of causation and paradox do not exist in the real world.  They make for extremely interestingly science fiction stories, but they are not going to happen in real life.