I am finding it difficult to understand the Calvinistic ideas of limited atonement and predestination. Does limited atonement mean that only a chosen people called the elect will attain salvation? If yes, will they all be Christians and/or non-Christians as well? How do I know that I am among the elect or not? If everything is predestined, then why preaching the Gospel of our Lord Jesus Christ is necessary? All these ideas just confuses me, please help, but from the Calvinistic Presbyterian point of view and a Biblical basis as well.
Reformed churches are confessional bodies so the best place to start would probably be with those confessions. There are generally two camps:
The English tradition uses the Westminster Confession of Faith: http://www.pcaac.org/resources/wcf/
The Continental tradition uses the 3 forms of unity - The Belgic Confession, The Canons of Dort and the Heidelberg Catechism: https://www.urcna.org/1651/custom/24288
I would recommend starting with the Heidelberg Catechism because its easier to understand. (It also has quite beautiful prose)
I will try to answer your specific questions using specific phrases from these confessions. The confessions are not authoritative and all of them contain bible references for what they teach.
(1) On the matter of Limited Atonement, Canons of Dort, article 9 states:
It is not the fault of the gospel, nor of Christ offered therein, nor of God, who calls men by the gospel and confers upon them various gifts, that those who are called by the ministry of the Word refuse to come and be converted. The fault lies in themselves, some of whom when called, regardless of their danger, reject the Word of life; others, though they receive it, suffer it not to make a lasting impression on their heart; therefore, their joy, arising only from a temporary faith, soon vanishes, and they fall away; while others choke the seed of the Word by perplexing cares and the pleasures of this world, and produce no fruit. This our Savior teaches in the parable of the sower (Matt. 13).
And article 10 states:
But that others who are called by the gospel obey the call and are converted is not to be ascribed to the proper exercise of free will, whereby one distinguishes himself above others equally furnished with grace sufficient for faith and conversion (as the proud heresy of Pelagius maintains); but it must be wholly ascribed to God, who, as He has chosen His own from eternity in Christ, so He calls them effectually in time, confers upon them faith and repentance, rescues them from the power of darkness, and translates them into the kingdom of His own Son
So yes, only a chosen people will attain salvation. For those that do, it is the work of God through his effectual grace. For those that do not, it is their choice to reject the gospel and the fault lies in themselves.
(2) Will they all be Christians? Yes. Heidelberg Catechism, question 20:
Q. ARE ALL MEN SAVED THROUGH CHRIST JUST AS ALL WERE LOST THROUGH ADAM?
A. No. Only those are saved who by true faith are grafted into Christ and accept all his blessings. Matt. 7:14; John 3:16, 18, 36; Rom. 11:16-21
Q. WHAT IS TRUE FAITH?
A. True faith is not only a knowledge and conviction that everything God reveals in his Word is true; it is also a deep-rooted assurance, created in me by the Holy Spirit through the gospel, that, out of sheer grace earned for us by Christ, not only others, but I too, have had my sins forgiven, have been made forever right with God, and have been granted salvation. John 17:3, 17; Heb. 11:1-3; James 2:19; Rom. 4:18-21; 5:1; 10:10; Heb. 4:14-16; Matt. 16:15-17; John 3:5; Acts 16:14; Rom. 1:16; 10:17; 1 Cor. 1:21; Rom. 3:21-26; Gal. 2:16; Eph. 2:8-10; Gal. 2:20; Rom. 1:17; Heb. 10:10
(3) How do you know you are among the elect?
If you declare with your mouth, "Jesus is Lord," and believe in your heart that God raised him from the dead, you will be saved.
Salvation is a work of God. Faith is believing in that work. If you believe, you will be saved.
Westminster Confession of Faith, chapter 18 says:
Although hypocrites and other unregenerate men may vainly deceive themselves with false hopes and carnal presumptions of being in the favor of God, and estate of salvationa (which hope of theirs shall perish) yet such as truly believe in the Lord Jesus, and love him in sincerity, endeavoring to walk in all good conscience before him, may, in this life, be certainly assured that they are in the state of grace, and may rejoice in the hope of the glory of God, which hope shall never make them ashamed. [...] This infallible assurance doth not so belong to the essence of faith, but that a true believer may wait long, and conflict with many difficulties, before he be partaker of it yet, being enabled by the Spirit to know the things which are freely given him of God, he may without extraordinary revelation, in the right use of ordinary means, attain thereunto [...] True believers may have the assurance of their salvation divers ways shaken, diminished, and intermitted; as, by negligence in preserving of it, by falling into some special sin which woundeth the conscience and grieveth the Spirit; by some sudden or vehement temptation, by God’s withdrawing the light of his countenance, and suffering even such as fear him to walk in darkness and to have no light yet are they never utterly destitute of that seed of God, and life of faith, that love of Christ and the brethren, that sincerity of heart, and conscience of duty, out of which, by the operation of the Spirit, this assurance may, in due time, be revived; and by the which, in the meantime, they are supported from utter despair.
Assurance is possible but sometimes hard. The ordinary means of grace being referred to here are things like regularly attending church. Much of what we do every Sunday is simply reminding us the promises of God because we are a forgetful people.
(4) If everything is predestined, then why preaching the Gospel of our Lord Jesus Christ is necessary?
God ordains the means as well as the ends and he has chosen the church as his vehicle of Salvation. Heidelberg Catechism, question 65:
Q. IT IS BY FAITH ALONE THAT WE SHARE IN CHRIST AND ALL HIS BLESSINGS: WHERE THEN DOES THAT FAITH COME FROM?
A. The Holy Spirit produces it in our hearts by the preaching of the holy gospel, and confirms it through our use of the holy sacraments.
Romans 10 again: (w/emphasis from me)
For there is no difference between Jew and Gentile—the same Lord is Lord of all and richly blesses all who call on him, for, “Everyone who calls on the name of the Lord will be saved.” How, then, can they call on the one they have not believed in? And how can they believe in the one of whom they have not heard? And how can they hear without someone preaching to them? And how can anyone preach unless they are sent? As it is written: “How beautiful are the feet of those who bring good news!”
The preaching of the Gospel is necessary for anyone to come to faith, but God ordains that as well!
Everything is done with the foreknowledge of God, and He uses and works through all things to bring about His purposes.
This means we act in free will and intend what we intend, but it is only God who brings the light of revelation and power. So in our perspective we take the steps that are before us, but they are there because God has called us.
We are not locked in or forced to do things. Think of Judas, who was called an apostle, saw all Jesus did yet betrayed Him to His death. There is a series of choices we make and compromises, which lead us to a walk and path.
Luke in acts puts it like this "When the Gentiles heard this, they were glad and honored the word of the Lord; and all who were appointed for eternal life believed." Acts 13:48
Without preaching no one hears, without the Holy Spirit working no one comes to faith. The apostles emphasised we know our calling by how our hearts and intention respond
Dear friends, let us love one another, for love comes from God. Everyone who loves has been born of God and knows God. Whoever does not love does not know God, because God is love. 1 John 4:7-8
Love like this is a choice and also part of Gods work within us. It does not come easily, and is something we need training in, but this is God heart.