Your quote describing predestinarianism is clearly referring to a certain interpretation of Romans 9. Romans 9, after all, uses similar verbiage:
You will say to me then, “Why [then] does he still find fault? For who
can oppose his will?” But who indeed are you, a human being, to talk
back to God? Will what is made say to its maker, “Why have you created
me so?” Or does not the potter have a right over the clay, to make out
of the same lump one vessel for a noble purpose and another for an
ignoble one? What if God, wishing to show his wrath and make known his
power, has endured with much patience the vessels of wrath made for
destruction? This was to make known the riches of his glory to the
vessels of mercy, which he has prepared previously for glory, namely,
us whom he has called, not only from the Jews but also from the
Here's what John Calvin says about the passage:
The reason why what is formed ought not to contend with its former,
is, that the former does nothing but what he has a right to do. By the
word power, he means not that the maker has strength to do according
to his will, but that this privilege rightly and justly belongs to
him. For he intends not to claim for God any arbitrary power but what
ought to be justly ascribed to him.
And further, bear this in mind, — that as the potter takes away
nothing from the clay, whatever form he may give it; so God takes away
nothing from man, in whatever condition he may create him. Only this
is to be remembered, that God is deprived of a portion of his honor,
except such an authority over men be conceded to him as to constitute
him the arbitrator of life and death.
He then argues thus, — There are vessels prepared for destruction,
that is, given up and appointed to destruction: they are also vessels
of wrath, that is, made and formed for this end, that they may be
examples of God’s vengeance and displeasure. If the Lord bears
patiently for a time with these, not destroying them at the first
moment, but deferring the judgment prepared for them, and this in
order to set forth the decisions of his severity, that others may be
terrified by so dreadful examples, and also to make known his power,
to exhibit which he makes them in various ways to serve; and, further,
that the amplitude of his mercy towards the elect may hence be more
fully known and more brightly shine forth; — what is there worthy of
being reprehended in this dispensation? But that he is silent as to
the reason, why they are vessels appointed to destruction, is no
matter of wonder. He indeed takes it as granted, according to what has
been already said, that the reason is hid in the secret and
inexplorable counsel of God; whose justice it behoves us rather to
adore than to scrutinize.
Though in the second clause he asserts more expressly that it is God
who prepares the elect for glory, as he had simply said before that
the reprobate are vessels prepared for destruction; there is yet no
doubt but that the preparation of both is connected with the secret
counsel of God. Paul might have otherwise said, that the reprobate
give up or cast themselves into destruction; but he intimates here,
that before they are born they are destined to their lot.
So in essence, yes, Calvin believed in an eternal decree to save some sinners and leave others to the destruction that their sin deserves.
To answer the second half of the question, Calvin believed that a Christian can have assurance of salvation, but warned that there are wrong ways to achieve such assurance. The first, as he writes in Instruction in Faith, is to look to the decree itself:
Let us take from the lot of both the elect and the others, reasons for
extolling his glory. On the other hand, let us not seek (as many do),
in order to confirm the certainty of our salvation, to penetrate the
very interior of heaven and to investigate what God from his eternity
has decided to do with us. That can only worry us with a miserable
distress and perturbation. Let us be content, then, with the testimony
by which he has sufficiently and amply confirmed to us this certainty.
Another wrong way is to look chiefly to oneself, as he describes in his Institutes of the Christian Religion (book 3, chapter 13, paragraph 13):
Let the most perfect man descend into his own conscience, and bring
his actions to account, and what will the result be? Will he feel calm
and quiescent, as if all matters were well arranged between himself
and God; or will he not rather be stung with dire torment, when he
sees that the ground of condemnation is within him if he be estimated
by his works? Conscience, when it beholds God, must either have sure
peace with his justice, or be beset by the terrors of hell. We gain
nothing, therefore, by discoursing of righteousness, unless we hold it
to be a righteousness stable enough to support our souls before the
tribunal of God. When the soul is able to appear intrepidly in the
presence of God, and receive his sentence without dismay, then only
let us know that we have found a righteousness that is not fictitious.
Calvin writes in a number of places that "the certainty of faith depends on the grace of Christ alone," and that, "We must begin with what is revealed in Christ concerning the love of the Father for us and what Christ Himself daily preaches to us through the Gospel." In other words, for assurance of faith we look to 1) the perfect obedience of Christ, 2) the mercy of God, and 3) the promises of salvation God gives to his church based on the first two items.
But in his commentary on 1 John 3:14, he says:
For as no one sincerely loves his brethren, except he is regenerated
by the Spirit of God, he hence rightly concludes that the Spirit of
God, who is life, dwells in all who love the brethren. But it would be
preposterous for any one to infer hence, that life is obtained by
love, since love is in order of time posterior to it.
The argument would be more plausible, were it said that love makes us
more certain of life: then confidence as to salvation would recumb on
works. But the answer to this is obvious; for though faith is
confirmed by all the graces of God as aids, yet it ceases not to have
its foundation in the mercy of God only. As for instance, when we
enjoy the light, we are certain that the sun shines; if the sun shines
on the place in which we are, we have a clearer view of it; but yet
when the visible rays do not come to us, we are satisfied that the sun
diffuses its brightness for our benefit. So when faith is founded on
Christ, some things may happen to assist it, still it rests on
Christ’s grace alone.
One way of reading this is that, to have confidence that we stand on the promises of God, we look chiefly to the strength of the foundation (Christ), but we can also to some degree look at whether we have built on that foundation via good works ("love of the brethren" chief among those works).
So to sum up: a person can have assurance that they themselves are elect by looking firstly unto Christ and his promises, then to one's works which are a fruit of saving faith. Assurance is not found by excessive introspection or by inquiring into the decree of election itself.
While I believe Calvin can accurately be stated to have "systematized" Calvinism, it received more systemization and scrutiny over the course of the following centuries. But, remarkably, the confessions of the reformed churches reached the same conclusions regarding assurance as he did.
All scriptural citations are from the NABRE. Though I prefer the Battles translation of the Institutes, I don't know where it can be found online so I quoted Beverige for ease of cross-reference. I am much indebted to the article Calvin's Doctrine of the Assurance of Faith by David B. McWilliams for a lot of the content of this answer.