"I
believe we (Christians) are chosen by God before He created the world and at any
point in time, a nonbeliever can become a believer (thus Christ gave the great
commission)."
-- There are a couple of things we need to keep straight in our thinking here (I also encourage you to carefully read the passages I provide as support):
1. As you rightly point out, election was something that God did, and He did this before He created the world.
2. As noted earlier, God elected persons, not merely a plan or a hypothetical entity for salvation. Those that God elected in Christ before time began will most certainly be saved in space-time history.
3. Though the elect were chosen in Christ before time began (Eph 1:3-6), and though they will most certainly be saved (John 10:27-28), the fact of the matter is that they are born into the world even as the rest: as sinners and unbelievers and children of wrath (Eph 2:1-3). So, it important now to keep in mind the point that God has ordained the end (the salvation of His elect), as well as the means to that end (the sending of Christ and preaching of the Gospel).
4. In terms of the means that God has ordained for the salvation of His elect, it was necessary that God send His Son, the second person of the Trinity, into the world to be the substitute for His people (Heb 2:9-18). So, Jesus had to take to Himself a human nature (Phil 2:6-8) , and He had to obey the law at all points (where we fail at all points) and had to suffer and die and pay the penalty for the sins of His people on the cross (Rom 5:19; 2 Cor 5:21). Then He had to rise from the grave for the justification of His people (Rom 4:25), and ascend into Heaven to intercede for His people (Heb 7:23 - 8:2). Then, in order for His people to be brought to faith, He has to bring people to saving faith (John 6:37;44), and they in turn must proclaim the Gospel to the whole world (i.e., the Great Commission) (Mark 16:15; Matt 28:18-20), and the reason for this is because the "elect" are found among the vast sea of people from every nation, tribe, and tongue (Acts 13:48; Rev 5:9). Faith comes by hearing, and hearing by the Word of God (Rom 10:17), and the Gospel is the power of God unto salvation (Rom 1:16), so, when the Word is proclaimed to all people, Christ's sheep, which are among the goats, will hear, and will be saved while the goats will not hear and will not be saved (John 10:25-29; Matt 25:31-34).
5. So, in conclusion of this point, we understand that election occurred before time began. But, we also understand that we do not know the identity of the elect; only God knows that. So, He has tasked us to go to all people proclaiming the Gospel, because the Gospel is the means through which God is going to bring His elect to Himself. So, keeping in mind the qualifications I've pointed out, your statement is true that an unbeliever can become a believer, and hence the Great Commission, because everyone, to include the elect, that is born into the world is born into sin and is a child of wrath; everyone is born as an unbeliever. But, only those that were marked out from all eternity as one of Christ's sheep will be saved, and this is solely by God's grace alone.