No, I believe that God takes the first step leading to salvation. And that grace is unconditionally offered whether the person accepts it or rejects it. Though the measure of grace may vary depending on God's foreknowledge (i.e. the Parable of the Talents).
Easy.
Paul puts 'believing/trusting' on the other side of the inequality between 'works of the law' and 'believing'. (Even though both are something you 'do').
4 Now to the one who works, wages are not credited as a gift but as an obligation. 5 However, to the one who does not work but trusts God who justifies the ungodly, their faith is credited as righteousness. - Romans 4:4-5
From his above statement it's impossible to say that 'believing' is on the same side of 'works' in regard to what doesn't justify.
It's really that simple. It plainly says that.
Some might not respond with believing/trusting to God's gracious gift of knowledge that the gospel really is true.
In fact, most won't.
#2 All persons without exception respond by believing or not believing. It's their choice.
Yes.
And they continue to believe through the exact same gift of faith (knowing) upon which they first believed.