John 10:26 - But ye believe not, because ye are not of my sheep, as I said unto you.
The reason they don't believe is that they are not His sheep (it does not say that they
could not believe, only that they
do not). This makes no reference to some ancient decree of election. The next verse defines His "sheep" (without the slightest hint of predestination): "My sheep hear my voice...and they follow me." Like the faithful remnant in Old Testament times, who listened to the prophets and obeyed their preaching, so also the remnant in Jesus' time heard and followed Him. This was happening as He spoke. If there was some prehistoric decree that determined that they would do so, Jesus is silent on it.
Acts 13:48 - And when the Gentiles heard this, they were glad, and glorified the word of the Lord: and as many as were ordained to eternal life believed.
This has been discussed at this forum many times and at great length. I also deal with it in my lectures on Calvinism. Suffice it to say, it does not tell us who it was that "ordained" or "disposed" these people to believe. Nor does it say that this disposition occurred at any time previous to their hearing Paul preach.
Acts 16:14 - And a certain woman named Lydia, a seller of purple, of the city of Thyatira, which worshipped God, heard us: whose heart the Lord opened, that she attended unto the things which were spoken of Paul.
Lydia was a worshipper of God before she met Paul and heard the gospel. She was part of the Jewish remnant (or else a faithful proselyte). The next step for her was to receive the gospel. Since she was one of God's own people already, it is not surprising that He opened her heart to listen to Paul. After she listened, the choice to believe or not was her own, as with the rest of us. Given her earlier disposition of being a worshipper, her acceptance of Paul's message was not surprising.
Acts 18:27 - And when he was disposed to pass into Achaia, the brethren wrote, exhorting the disciples to receive him: who, when he was come, helped them much which had believed through grace
They "believed through grace," just like the rest of us. I believe that the grace of God is prevenient to any approach to Him. This grace may take the form of providential opportunities to hear, and the intellectual capacity to understand the gospel correctly. Previous graces, such as conviction of sin, special benevolences received and appreciated, or other circumstances that might favorably dispose a person to respond favorably often may also be factors. Every Christian must surely acknowledge that God has shown great grace to us, which preceded our coming to Him, and somewhat influenced our disposition to respond.
II Thessalonians 2:13 - But we are bound to give thanks alway to God for you, brethren beloved of the Lord, because God hath from the beginning chosen you to salvation through sanctification of the Spirit and belief of the truth
"You" means the believing community. The church (collectively) has been chosen by God to obtain salvation. How? Through the sanctifying of the Spirit and through belief in the truth. This is what salvation involves. It is not clear what is meant by "from the beginning." It may mean that God chose before the foundation of the world that He would save "in Christ" those who would come to Him (as in Eph.1:4; 2 Tim.1:9; Titus 1:2), or "from the beginning" might mean "early in my ministry," or "early in the overall gospel era." It is not clear what the referent may be to the word "beginning."
I Corinthians 2:14 - But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know them, because they are spiritually discerned.
The "things of the Spirit of God" is not a reference to the gospel. Unbelievers actually do receive the gospel on a regular basis around the world (and Paul doesn't say "can't" but "do not"). Paul is not talking about people getting saved. The context tells us that Paul is talking about the "deep things of God" (v.10), which he teaches "to the mature" (v.6), but which he was unable to teach the carnal, immature Corinthians (3:1-3). It is the Corinthian Christians (not unbelievers) whom he says were "not able to receive" these things (3:3). There is no indication that unbelievers cannot receive the gospel. The Corinthians (though believers) were unable to receive the "things of the Spirit" of which Paul was speaking. However, they had, in fact, as unbelievers, been able to receive "Jesus Christ and Him crucified" (2:2).
). Even concise answers like the ones above are great because they can serve as a branching off point for me and I can dig in a little deeper in my own studying.