Sin is not the consequence of a hardened heart, it is the natural disposition of mankind after the fall. The hardening induced by God is related to the response to the opportunity for repentance and faith.
Let no man say when he is tempted, I am tempted of God: for God cannot be tempted with evil, neither tempteth he any man: But every man is tempted, when he is drawn away of his own lust, and enticed. Then when lust hath conceived, it bringeth forth sin: and sin, when it is finished, bringeth forth death. (Jas 1:13-15)
So every sin committed is the result of your own lusts that your did not overcome:
If thou doest well, shalt thou not be accepted? and if thou doest not well, sin lieth at the door. And unto thee shall be his desire, and thou shalt rule over him. (Gen 4:7)
The final analysis, however, is that no-one has succeeded in this:
As it is written, There is none righteous, no, not one: There is none that understandeth, there is none that seeketh after God. They are all gone out of the way, they are together become unprofitable; there is none that doeth good, no, not one.
For all have sinned, and come short of the glory of God; (Rom 3:23)
And the problem is amplified: not just did they not overcome their own lusts, they cannot - sin is in their flesh making man incapable of obedience, even if they loved God's law:
For the good that I would I do not: but the evil which I would not, that I do. Now if I do that I would not, it is no more I that do it, but sin that dwelleth in me. I find then a law, that, when I would do good, evil is present with me. For I delight in the law of God after the inward man: But I see another law in my members, warring against the law of my mind, and bringing me into captivity to the law of sin which is in my members.
So everyone is guilty already (Rom 3), unable to afford the cost of his guilt and unable to overcome the law of sin (Rom 7) in his own flesh that is continuing to compounded sins against him.
O wretched man that I am! who shall deliver me from the body of this death? I thank God through Jesus Christ our Lord. So then with the mind I myself serve the law of God; but with the flesh the law of sin.
The solution is justification (Rom 4) and regeneration (Rom 8). Justification and regeneration are both components of salvation that is a free gift (grace) obtained through faith.
- Justification is the declaration of innocence obtained by faith alone through the substitutionary atonement of Christ on the cross,
For all have sinned, and come short of the glory of God; Being justified freely by his grace through the redemption that is in Christ Jesus: Whom God hath set forth to be a propitiation through faith in his blood, to declare his righteousness for the remission of sins that are past, through the forbearance of God; To declare, I say, at this time his righteousness: that he might be just, and the justifier of him which believeth in Jesus.
- While regeneration is the deliverance from the Law of Sin and death in our flesh.
O wretched man that I am! who shall deliver me from the body of this death? I thank God through Jesus Christ our Lord. So then with the mind I myself serve the law of God; but with the flesh the law of sin. There is therefore now no condemnation to them which are in Christ Jesus, who walk not after the flesh, but after the Spirit. For the law of the Spirit of life in Christ Jesus hath made me free from the law of sin and death. (Rom 7:24-8:2)
This Spirit is also received by faith:
That the blessing of Abraham might come on the Gentiles through Jesus Christ; that we might receive the promise of the Spirit through faith. (Gal 3:14)
That means that all are in the same burning sinking boat (set alight by themselves) offered the same offer of escape through faith... so why do some take it and others not take it?
Because faith itself is a gift...
For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: (Eph 2:8)
For unto you it is given in the behalf of Christ, not only to believe on him, but also to suffer for his sake; (Php 1:29)
So if two people have an opportunity to hear, repent and believe the Gospel, then what is the difference between them if one receives it and the other rejects it?
One was shown mercy through the gifts of faith, and the other was hardened in his through his unrepentance.
Romans 9 which employes the explanation of hardenings, comes after the offerand explanation of Justification and Regeneration to defend any accusation why some (especially among Israel) may not be finally saved - because they reject the offer... by the hardening induced by God. Why? Paul seems to simply offer a possible explanation:
What if God, willing to shew his wrath, and to make his power known, endured with much longsuffering the vessels of wrath fitted to destruction: And that he might make known the riches of his glory on the vessels of mercy, which he had afore prepared unto glory, (Rom 9:22-23)
The point in answer to the OP's question is that sin is not the consequence of the hardening, it pre-exists the offer of forgiveness and redemption, and hence the response of faith or hardening.
So in answer of the OP's two questions:
- No, God does not override the individual's free will, he is a slave to sin that is in his flesh, his will is not free - it is subject to his own ability (or rather inability) to overcome his own lusts. God empowers those on whom he shows mercy through the gift of faith, with the ability to freely choose salvation.
So no one is guilty of sin because God hardened their hearts, but everyone that has their sins forgiving can definitely blame God for obtaining mercy.
- Irrelevant since it depends on the answer 1 being yes.
To answer the overall accusation, that God is responsible for sin: it would be more accurate to say God ordains it and uses sin as He uses everything, for His purposes and glory.
Here are a few examples:
- God used Joseph's brothers' sin for his purposes:
But as for you, ye thought evil against me; but God meant it unto good, to bring to pass, as it is this day, to save much people alive. (Gen 50:20)
- God hardened Pharoah's heart to make his power known:
And in very deed for this cause have I raised thee up, for to shew in thee my power; and that my name may be declared throughout all the earth. (Exo 9:16)
- God hardened the heart of the Canaanite nations who could have made peace with the invading Israel, in order that God may uphold His word to Moses. If a nation made peace, they would be spared, but God commanded them to destroy all these nations, and hence hardened their hearts.
There was not a city that made peace with the children of Israel, save the Hivites the inhabitants of Gibeon: all other they took in battle. For it was of the LORD to harden their hearts, that they should come against Israel in battle, that he might destroy them utterly, and that they might have no favour, but that he might destroy them, as the LORD commanded Moses. (Jos 11:19-20)
But of the cities of these people, which the LORD thy God doth give thee for an inheritance, thou shalt save alive nothing that breatheth: But thou shalt utterly destroy them; namely, the Hittites, and the Amorites, the Canaanites, and the Perizzites, the Hivites, and the Jebusites; as the LORD thy God hath commanded thee: (Deu 20:16-17)
- Israel, however, did not obey this instruction and even this appears was God's ordained will to intentionally leave some of these nations which they were commanded to destroy to eventually prove Israel - and they were continuously found unfaithful...
I also will not henceforth drive out any from before them of the nations which Joshua left when he died: That through them, I may prove Israel, whether they will keep the way of the LORD to walk therein, as their fathers did keep it, or not. Therefore the LORD left those nations, without driving them out hastily; neither delivered he them into the hand of Joshua. (Jdg 2:21-23)
- God used the adulterous and murderous acts of David, to produce a linage for the Messiah:
And Jesse begat David the king; and David the king begat Solomon of her that had been the wife of Urias; (Mat 1:6)
- God uses the unbelief and murderous actions of the Jewish and Gentile leaders to ultimately accomplish his great work of Salvation
Him, being delivered by the determinate counsel and foreknowledge of God, ye have taken, and by wicked hands have crucified and slain: (Act 2:23)
For of a truth against thy holy child Jesus, whom thou hast anointed, both Herod, and Pontius Pilate, with the Gentiles, and the people of Israel, were gathered together, For to do whatsoever thy hand and thy counsel determined before to be done. (Act 4:27-28)