How can one refute the pre-millenialist position? Does it mean that we are given a second chance at salvation?
To my way of thinking, the easiest way to refute the premillennialist position is to look at the scriptures they use in their context. I will look at some of them later. But first I will answer your second question.
There are several premillennialist positions regarding what they call the "rapture." Some say it has happened, others (most?) that it is yet to happen. The typical scenario, as I understand it, is that Jesus will come and "rapture" (catch up, snatch) the righteous believers. Those that are left will have seven (or three and a half) years to obey before the millennium (or judgement) in which to repent. Those who are caught up don't get, or need, a second chance. Those that died prior to that time don't get a second chance. If there is a "second chance at salvation" it would be for those who were alive at the time of the rapture. Although most try to say there is no second chance, they also say that seven year period is one in which those who were alive may repent before the end. To me this sounds like God will arbitrarily give some people a second chance that others did not have. Those left behind will, theoretically, be aware of the disappearance of all the believers. They will still have the word of God. They will have the ability to repent, although some say nobody in that period will do so. Unless they have no free will in the matter, yes, this scenario means that some will be given a second chance at salvation. I know of no direct scriptures that say people can not be given a second chance. The scriptures I would use to refute this part of the doctrine all relate to God being a just God. Arbitrarily offering some an opportunity that all those who preceded us did not have is hardly a definition of justice.
Now to look at some scriptures that are used to support premillennialism. Some parts of the doctrine have no scriptural basis. For instance, the Bible never refers to seven years of tribulation. (See my article "Is This the Beginning?") Some parts are based on symbols in prophetic works. Unfortunately, many ignore one basic rule of interpretation of prophecy or parables, that you must not take each symbol and overanalyze them rather than looking at the overall meaning. Some doctrines rest on wresting some scriptures to fit with an interpretation of other scriptures.
First let's look at 1 Thes 4:15-17. This is, to me, the one of the biggest refutations of premillennialism. The passage says that the dead will not be worse off than the living when Jesus comes. Instead, Jesus will come with great noise, the dead in Christ (indeed all dead) will be raised, the living will be caught up with them, we will meet Jesus in the clouds and forever be with him. This says that the living and dead will all rise at the same time. Additionally, the passage implies that Jesus will not set foot on earth again.
One part of the premillennial doctrine is that Jesus will rule on David's throne here on earth. This is in direct opposition to his statement, "my kingdom is not of this world." (Jn 18:36) It is also in opposition to Jeremiah 22:28-29, which says that no descendent of King Jeconiah will rule from the throne of David, which would include Jesus. (See my article, "Jesus Genealogy".)
The basic doctrine of this position is that Jesus will reign (or is reigning) for 1000 years in a kingdom on this earth. Of course, any time prophecy speaks of a thousand years, it is not to be taken literally, but simply representative of a long period of time, just as the word "north" meant any invader of Israel from anywhere besides Egypt. Another thing to notice is that this seems to be saying that when Jesus came the first time he was unable to set up his kingdom like God had originally planned. (See my article, "Replacement Theory") To me, though, the most important scripture about this is 1 Cor 15:20-26. This says that Jesus will give up the kingdom after the resurrection of the dead. Since the passage indicates that this will happen immediately after the resurrection (vv. 25-26) rather than 1000 years later as some would say, then Jesus must be reigning in a kingdom before he comes again. This means he is reigning now, and has done so for almost 2000 years. If he is king of his kingdom now, why would he want to demote himself to be king of a physical, limited earthly kingdom?
Much of the doctrine comes from a combination of the book of Revelation (which was to happen shortly after it was written-Rev 1:1) and chapters 24 and 25 of Matthew. But in Matthew 24, Jesus clearly says that those who say the Messiah has come again because of wars, disasters, and tribulation are not telling the truth. Instead, he says, his coming and the end of the world will be without warning. Matthew 25 says that he will judge the world at that time, not a thousand years later. The book of Revelation is clearly speaking about Christian persecution under the Roman Empire. To put it in a time future to us (or even contemporary to us) requires that we put modern interpretations to first century symbols (an illogical act) and that we ignore John's statement that these events would happen to those of his day.
Your questions do not have simple answers. I hope I have given you something to start with. Just remember that the easiest way to refute the doctrine is to look at the scriptures they use, in context.
Finally, I would point out that the doctrine of premillennialism does not speak to anything necessary for salvation. If it should prove to be true, which would mean that scriptures contradict themselves, that would not change the fact that those who would come to God must repent, be immersed in water, and rise to walk in obedience.