According to the Catholic Church, does Jesus condemn those who don't believe?

It is not as simple as a yes or no. The relevant section of the Cathecism says the following are important in understanding the statement "Outside the Church there is no salvation":

• "salvation comes from Christ the Head through the Church which is his Body" - this doctrine, as all doctrines of the Church, is centered on Christ. It is not about punishing people, looking down on others, etc.

• "This affirmation is not aimed at those who, through no fault of their own, do not know Christ and his Church. Those who, through no fault of their own, do not know the Gospel of Christ or his Church, but who nevertheless seek God with a sincere heart, and, moved by grace, try in their actions to do his will as they know it through the dictates of their conscience - those too may achieve eternal salvation." - Note though, this is not simply excusing those who don't know any better.

• "the Church still has the obligation and also the sacred right to evangelize all men" - again, the point is not that we're lucky because we have salvation and we should look down on others who don't have salvation, rather we have a duty by virtue of our Baptism to evangelize.

The Haydock Commentary on that verse says:

Ver. 47. I do not judge him. To judge here, may signify to condemn. St. Augustine expounds it in this manner: I do not judge him at this my first coming. St. Chrysostom says, it is not I only that judgeth him, but the works also that I do.

Thus, during His 1st coming He doesn't judge, but he certainly will at His 2nd coming.

I think if you observe your quotes closely, your answer is revealed within them. Notice in the John passage Jesus states, for I did not come to condemn the world. When was it that he had said this, it was in his incarnation, his first arrival to earth, to carry out his salvific act, the act that would allow reconciliation to all who would place their trust in him.

Now there are two stipulations in that passage which carry you into the second part of your answer. First, this condemnation will not be extended to those who accepted the words that he spoke (all of his words which pointed to him being the source of this salvation); and second, this condemnation for those who have rejected his words (that which stands as a witness to what he has spoken) will be applied on the last day, his second coming, which correlates with what the The Nicene Creed says.

In summary, the grace period is extended from his first visit to his second, this is the time of mercy when no one is judged. Though this judgment comes from his own lips, those words that testified to who he was, from the time of his first arrival, will not be applied till that last day, his second coming, when they are used as a witness, testifying in judgement upon those who did not accept them.

People are not condemned, judged, or punished because they do not have faith in Jesus. They are condemned because of the evil acts they have committed during their life on Earth. They are saved from punishment by faith in Jesus, and (depending on your view on justification by faith alone or with works) by those works of righteousness that spring from that faith. Lack of faith leads to lack of forgiveness and justification, but that forgiveness and justification is only needed because of the sins already committed. As an analogy, the governor of a state by statute may pardon a criminal on death row. If he does not pardon the criminal, and the state has capital punishment, you could say that the governor condemned him to die, but it is really the crimes that the person committed that really condemned them.

Matthew 12:

36 I tell you that on the day of judgment people will have to account for every careless word they speak. 37 For by your words you will be acquitted, and by your words you will be condemned.”

John 3:16

“For God loved the world in this way: He gave His One and Only Son, so that everyone who believes in Him will not perish but have eternal life. 17 For God did not send His Son into the world that He might condemn the world, but that the world might be saved through Him. 18 Anyone who believes in Him is not condemned, but anyone who does not believe is already condemned, because he has not believed in the name of the One and Only Son of God.

19 “This, then, is the judgment: The light has come into the world, and people loved darkness rather than the light because their deeds were evil. 20 For everyone who practices wicked things hates the light and avoids it, so that his deeds may not be exposed. 21 But anyone who lives by the truth comes to the light, so that his works may be shown to be accomplished by God.”

John 7 shows the clearest connection between faith and forgiveness, where a sinful woman demonstrates her faith through her loving act of anointing Jesus:

36 Then one of the Pharisees invited Him to eat with him. He entered the Pharisee’s house and reclined at the table. 37 And a woman in the town who was a sinner found out that Jesus was reclining at the table in the Pharisee’s house. She brought an alabaster jar of fragrant oil 38 and stood behind Him at His feet, weeping, and began to wash His feet with her tears. She wiped His feet with the hair of her head, kissing them and anointing them with the fragrant oil.

39 When the Pharisee who had invited Him saw this, he said to himself, “This man, if He were a prophet, would know who and what kind of woman this is who is touching Him—she’s a sinner!”

40 Jesus replied to him, “Simon, I have something to say to you.”

“Teacher,” he said, “say it.”

41 “A creditor had two debtors. One owed 500 denarii, and the other 50. 42 Since they could not pay it back, he graciously forgave them both. So, which of them will love him more?”

43 Simon answered, “I suppose the one he forgave more.”

“You have judged correctly,” He told him. 44 Turning to the woman, He said to Simon, “Do you see this woman? I entered your house; you gave Me no water for My feet, but she, with her tears, has washed My feet and wiped them with her hair. 45 You gave Me no kiss, but she hasn’t stopped kissing My feet since I came in. 46 You didn’t anoint My head with olive oil, but she has anointed My feet with fragrant oil. 47 Therefore I tell you, her many sins have been forgiven; that’s why[n] she loved much. But the one who is forgiven little, loves little.” 48 Then He said to her, “Your sins are forgiven.”

49 Those who were at the table with Him began to say among themselves, “Who is this man who even forgives sins?”

50 And He said to the woman, “Your faith has saved you. Go in peace.”