If I may, I'd like to respond to each of the passages you cited. As a Calvinist, I interpret them differently than you do. Maybe I'm wrong, in which case I am open to correction. So far, however, I've not encountered a rival interpretation that does justice to the whole counsel of Scripture. It seems there is always a passage or two elsewhere that is sacrificed in the process. (Imagine solving one side of a Rubik's Cube at the expense of two others.)
I will provide numerous scriptures in support of my interpretations, and I would strongly encourage you to look them up as you go. If you feel that I'm ripping any out of context, please let me know.
1.) JOHN 3:16 -- "For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life."
There are two things that the student of Scripture should keep in-mind, here. The first (as usual) is context. In the previous two verses, Jesus explained to Nicodemus that "Just as Moses lifted up the snake in the desert, so the Son of Man must be lifted up, that everyone who believes in him may have eternal life." Prior to the New Covenant, God dealt mainly with the people of Israel. When Moses lifted the serpent, it promised healing for those Hebrews who looked upon it in faith. Christ, in contrast, would provide healing not only for the Jewish people but for the Gentiles as well. That was, after all, the original plan. Through the seed of Abraham--the Father of the Jews--all nations of the earth would be blessed (Gen 22:18). Christ is that seed, and those who have faith in him are indeed Abraham's children (Gal 3:29). Therefore, Christ's statement about God's love for the world must be understood in light of Gentile inclusivity. It's not that God loves every single individual in the world without exception. Rather, that His love is now inclusive of all nations; it is not confined to ethnic Israel (cf. 1 John 2:2). And just as God had chosen a remnant among the Jews (Rom 11:1-5), so too does He choose among the Gentiles (Eph 1:3-6; 1 Pet 2:9-10).
Second, there is nothing in this verse to suggest that all men may believe in Christ. True, English renderings like the one above may give that impression. Interestingly, the word 'whoever' does not appear in the original text. The Greek phrase is pas ho pisteuon, which literally means "all the believing ones." Whereas the English is somewhat vague and open-ended, the Greek is not. Jesus died for "all the believing ones." Scripture refers to this group as the elect (2 Tim 2:10; cf. Acts 13:48).
As for man's ability to come to Christ, we already noted that John 3:16 is silent on that matter. Thankfully, we don't have to look far beyond it for a treatment of man's ability (or rather, inability) to believe. In the very same chapter, Jesus told Nicodemus that a person must be born again (Gk. anothen, literally "born from above") in order to see and enter the Kingom of God. Only three chapters later, we find a case in point. Some disciples who were offended by Jesus' teaching had abandoned him. Jesus offers this explanation: "The Spirit gives life; the flesh counts for nothing. The words I have spoken to you are spirit and they are life. Yet there are some of you who do not believe... This is why I told you that no one can come to me unless the Father has enabled him" (6:63-65). So you see, human beings are incapable of coming to Christ and believing in him until the Father first enables them to do so. Obviously, he doesn't enable everybody, otherwise Jesus' explanation would be in error and the point moot.
2.) 2 PETER 3:9 -- "The Lord is not slow in keeping his promise, as some understand slowness. He is patient with you, not wanting anyone to perish, but everyone to come to repentance."
Here, context is also very important. When Peter writes, "He is patient with you," to whom is he speaking? The introduction of the letter answers that question. "To those who through the righteousness of our God and Savior Jesus Christ have received a faith as precious as ours:" (1:1b). Clearly, Peter is writing to Christians. Specifically, he is writing to God's elect, for in the same chapter he exhorts them to make their "calling and election sure" (v. 10).
In what way is God patient with His elect? Again, the context reveals that Peter is speaking about the day of Judgment (2:10). God is postponing the judgment, not because He is slow, but because all of the elect have not yet come to faith (e.g., John 10:16). If the Day of the Lord were to come before the elect were brought into the fold, they would be forever lost. And this cannot possibly happen, for Christ shall lose none of all that the Father has given to him (John 6:39).
Thus, in light of the above, Peter is essentially saying: "God is patient with you [the elect], not wanting anyone [of the elect] to perish, but everyone [of the elect] to come to repentence." As Paul writes elsewhere, "everything for the sake of the elect" (2 Tim 2:10).
3.) 1 TIMOTHY 2:3-6 -- "This is good, and pleases God our Savior, who wants all men to be saved and to come to a knowledge of the truth. For there is one God and one mediator between God and men, the man Christ Jesus, who gave himself as a ransom for all men—the testimony given in its proper time."
Naturally, the first question one should ask is, "What is good, and pleases God our Savior?" That requires us to back-up a step. "I urge, then, first of all, that requests, prayers, intercession and thanksgiving be made for everyone—for kings and all those in authority, that we may live peaceful and quiet lives in all godliness and holiness." Prayer for the poor and needy comes naturally. Prayer for rulers and those in authority--especially when such rulers are persecuting you--is decidedly unnatural. And so Paul specifically requests prayer for that class of people. The gospel isn't meant for the poor and oppressed only. God intends to save all kinds of people, from paupers to princes. Thus, the Apostle's words in verse 4 should be interpreted as all men without distinction, not all men without exception.
The above is a typical Calvinist interpretation of the passages cited. I will neither be surprised nor offended if you happen to disagree. Hopefully, you will at least grant I have treated the text fairly and thoughtfully. By all means, correct whatever you perceive to be misinterpretations of the word.