"For from him and through him and to him are all things. To him be the glory forever. Amen".
Short answer: Yes God gives and withholds grace, He gives or withholds hearts to love Him and heaven and hell are the outcomes of how He made us. All things are from Him.
Long answer: You use the word "choice" in your question but you do not define
it. Can we define our terms?
The Bible uses the word "choose/choice". Joshua 24:15 "choose for yourselves whom you will serve". But what does it mean?
When someone "chooses" to praise God because the love of God is ruling in their life then yes , they make a choice in that they follow a particular pattern of behaviour, but is it them or a force beyond themselves that is carrying them along?
When someone "chooses" to "obey unrighteousness" [Romans 2:8] they also may be said to be making a "choice".
For both of the above scenarios I have used the word "choice", but I have not differentiated between "free choice" and "determined choice".
One definition of "free choice" might be: "Free will is free". Free of cause and effect; free of logic and rational. Being free of everything which might tie it down it is random and meaningless. Even if it selects the "right" answer that would be a fluke and therefore meaningless.
"Determined choice": "I am choosing what to write now" but it is only an outworking of how God made me. We choose; we choose on the basis of who we are; we do not choose who we are because we do not create ourselves. Our choices reflect how God made us. [I didn't choose the DNA or circumstances that I was born into, or when I was born again or that I was born again. John 3:7].
The trouble with "determined choice" is that some [e.g. Loraine Boettner and other compatibilists] think that if God was ultimately responsible for evil that that would make God evil. But we can ask if that is true.
If God, who is holy and only ever holy [1 Peter 1:15-16] creates something which is not himself and proves to that thing that it is not perfect of itself, that the created thing is not all knowing; that it is not all powerful; that it is not self-dependant; that it is not eternal; that it is not Him. Then if God can do that for a holy motive it means that God can sustain the existence of Adam when Adam fell without compromising His own holiness.
"All things were made through Him, and without Him was nothing made that was made".
Adam was made by God. If there is a exact relationship between how God made Adam and Adam's behaviour, then God is ultimately responsible for Adam's behaviour. If no relationship exists then Adam is free, but free of cause and effect, free to behave in a way which denies how he was made and who he is, that denies the nature he was given.
I have tried to look at some of the issues that lie behind "choose", "ultimate responsibility" and God giving and withholding. These sorts of discussions have been around for a long time.
"And it shall come to pass in the last days, says God, That I will pour out My Spirit on all flesh;".
God withholds before the last days, but when He does give His Spirit, people are given a heart to know Him.