First of all, I don't know your meaning of "deletion" of a passage, so I'll try to give you some bits of general information.
Bible is a work of group of people, it's a tradition, not a text "downloaded" straight from God to Earth.
Even when we talk about a work of one person (Gospel of XX, Book of XX), it's really not. For example, what we call "Paul's Epistle to ..." is not actually written by Paul - he only dictated the message to his clerk (there is even evidence of this in Romans 16:22 "I, Tertius, who wrote down this letter, greet you in the Lord", and in other place, he explicitly states he has the urge to write a part of episle himself, because of the circumstances - Galatians 6:11 "See what large letters I use as I write to you with my own hand!"). Naturally, the person who wrote it down could have made a mistake or change something. He was human, humans can fail.
But there is also a possibility someone composed the message and signed it as Paul, in order to give it more weight - this is considered about 2nd Thessalonians, because the style and beliefs expressed concerning the second coming are very different to 1st Thessalonians. Not only could that be done in order to give weight to your word, but generally, signing your work with the name of your master/someone you respect as a teacher was common in ancient times. For example, hardly any manuscripts signed by Socrates really belong to him, same can be said for Pythagoras, Hyppocratus, etc.
But then, even after it's written, you had only one copy of the written text on your hand, but you'd want to spread it, so you had to make duplicates. They didn't have direct ways of copying texts back then like we do now - the printing press was invented way way later into the middle ages, so, people re-wrote texts to make copies! Obviously, when someone re-writing a text saw a mistake (as he thought) or a sticking out fragment, he would sometimes correct it. For example, Mark 1 "41 Jesus was indignant." and "43 Jesus sent him away at once with a strong warning: " - if I recall correctly, in original greek manuscript in one of these verses a verb "to roar like a lion" is used to express Jesus reaction towards the man. Obviously, those rewriting the texts wanted to soothe this passage, because they didn't feel that it fit for Jesus to make such expressions. Eventually, 1:41 turned into a feeling of compassion for the sick man.
Also, some churches or groups of people could have lost parts of the whole text, some could have bundled together texts that they thought were part of one whole, while they weren't (read about 2nd Corinthians, and how it is considered to contain several different missives all in one - for example, here http://www.earlychristianwritings.com/2corinthians.html)
There are certain verses that may likely have been added at a certain point (meaning, they were not part of the original manuscript written by original author in the first go). A big example would be the whole ending of the Gospel by Mark (16:9-20). The last chapter of Gospel by John is also likely an addition (but by John himself) to the originally complete text (it is obvious from the chapter's contents why the addition was made). There are also single verses added here and there, or "Amen" added to the end of several Paul messages that originally wasn't there. And so on.
You have to understand: there is no problem with that, because this isn't "falsification" or "alteration" of the Bible, as those who attack the Word of God would like to believe, when they say "Bible was altered many times and thus it's authenticity is questionable". Nothing essential, nothing "false" was added. And nothing significant was lost. Sure, it would be great to have all the Epistles to Corinth by Paul (which there were at least four, the one before the one we know as 1st, and the one between the ones we know as 1st and 2nd). But we have what we have. And we believe that since God did not make effort to preserve the rest, these are enough for us.
So, yes, to conclude: certain verses (or even whole bunch of verses) in the Bible could (or most likely were) added later down the line. And you could consider "deleting" them (which is done in some modern translations, where those verses are not present - for example, does your translation have verses 44 and 47 of Mark 9?)
Should you do so? Consider the fact I started with. Bible is a tradition, a work of group of people. The fact that you consider New Testament to contain the books it contains, in the form in which they are there - this was not a result of direct Jesus's will, or any of the apostles. It happened naturally, through the work of the Holy Spirit, and was finalised officially only in ~690. You can read about it here http://en.wikipedia.org/wiki/Development_of_the_New_Testament_canon So, if naturally some verses were added, and got accepted by the vast majority of the Christians, why consider them "to be ignored or deleted" just because the original author in the original manuscript didn not have them?
This is definetly not the case when you don't like the verse personally and want it to "not be there". This is basically trying to twist the words of God to fit your needs. People want to remove verses from the Bible every day - like verses about homosexuality, because they want Christianity to be fit to their liking.
I hope I need not explain why this is not a good idea.
Therefore, you are right in your second point - since we obviously cannot ignore and delete anything from the Bible because we don't understand it or don't like it, we should make effort to understand it to the best of our abilities.
On topic of actual meaning of the verse:
1) First of all, it literally means what it means.
Jesus warned people not to be fooled, because his teaching would not bring a way to make everybody live in peace, because that is impossible (because there are people who do not wish to do so, and God does not control the will of men). He warned people that his religion would divide even the most hard bounds - those of the blood - father and son, mother and daughter (which meant way more to the people of his time and in the area where he preached, than it does today to us). As it did (this was one of the reason for prosecution of Christianity by authorities). He obviously didn't want to "bait and switch", so he warns his disciples beforehead, that chosing to go with Him will have very serious consequences they should be aware of BEFORE they make the choice.
If you study the flow of the narrative of the Gospels, you will notice that the closer Jesus is to crucifiction, the more "serious" his preaching becomes. Because he saw that people misunderstood Him. They considered Him to be a miracle worker, a healer, even Messiah (meaning, a person who'll become a king like David, smash the idolaters and conquer the world for Jews). Even his disciples misunderstood Him. He obviously saw that and tried to correct it. Hence such "unpleasant" verses. Hence frequent mention of how disciples did not understand what he said, how he told them to be silent about him being the Messiah, about him curing people, etc.
2) Second, it always helps to read relative places in the Bible. This passage corellates with many others in the Bible. Here's the immediate two:
* Luke 14:26 If anyone comes to me and does not hate father and mother, wife and children, brothers and sisters—yes, even their own life—such a person cannot be my disciple.
* Matthew 10:37 Anyone who loves their father or mother more than me is not worthy of me; anyone who loves their son or daughter more than me is not worthy of me.
I'm sure there are others too. This will let you feel the overall picture. And also show you why you can't consider one single verse to be out of it's place, can't ignore or delete it - because it isn't out of it's place, it's very well in it's place and it corellates with other things Jesus had to say.
3) How to apply it to your own life?
Well obviously, this verse tells us that following after Christ can get in the way of our family life, our relationships with our relatives, and in that case, following Christ will have to take precedence. We all hope this will not go down to such levels, but sometimes, it will, and in this case, we have been forewarned that we will have to take a side - Jesus or my father/mother/relatives. Notice how there's nothing said about division of wife and husband, by the way.
I'd love to cite you good commentaries for the verse (and relevant verses) but I'm russian and all good commentaries I know are in russian too. I guess you can easilly find those for yourself in your language of preference if you google (for such a controversal verses, there must be many available). Read multiple ones and find out what explains the verse better to you. In the end, it's not about finding "true meaning" or "true interpretation" - it's about the Word of God being beneficial to you, influencing your life, helping you become a better person.
There is no "true" and "only" interpretation. There can only be false ones - those that twist the meaning, obviously, and should be disregarded as such (like muslims saying that John 16:13-15 talks about Mohhamed). But from those that are not false, you are free to peruse the ones that you feel resonate with yourself.