The answer to your question must be answered by first making some assumptions about his wife and children. Before you can do that, you have to make up your mind about who Jesus is because that will inform the kind of woman he would have married and how his children would have been raised.
First off, Jesus was a real person - let's just get that out of the way. There are four books in the Bible detailing his birth and later life. These books are corruburated by external sources, regarding the time and location of Jesus of Nazareth, so it's not just a "Bible told me so" claim.
Second, the majority of the information that we have on Jesus comes from the four Gospel accounts - Matthew, Mark, Luke and John - and only three of these were disciple of Jesus. Luke was a physician that traveled with Paul the apostle. These accounts of Jesus life seem fantastic to most and leave many with the opinion that Jesus was a peace loving, wise man. But that's because they've never really read the accounts.
Jesus did love peace, but he understood that is was not possible and that he, himself, was a lightning rod for controversy. He constantly challenged religious authority and the hypocritical.
He was wise, but the texts all indicate that he displayed a wisdom that was counter intuitive to the day. His teachings weren't just the Levitical law, but went past the law to the heart and held his listeners accountable for what they thought, not just what they did. In the end this rubbed many the wrong way.
Finally, the texts about Jesus detail miraculous signs - changing water into wine, healing the sick, feeding thousands with very little food at hand, walking on water, calming storms, raising the dead. In the end the accounts tell us he was killed, as phrophesied, and then rose from the dead to atone for the sins of man and conquer death. He claimed this power was given him by God and that he was equal with God. This claim alone should cause one of two responses.
- Wise men don't make claims of deity and he was a raving lunatic
- He was who he claimed to be and there are implications to this claim for any that believe it
Which ever you decide, your story should follow suit.
Jesus had a mother and a father and several brothers and sisters. Of these, only Mary, his mother, was ever honored to a place of semi-worship. Joseph, his father, is seemingly only notable in the nativity stories and none of his siblings is ever mentioned, except for James and Jude, who authored letters to the Jews in the New Testament of the Bible.
I mention all of this, because a relationship to Jesus did not make for automatic fame for most of the rest of his family. In my opinion it wouldn't have made his wife and children famous either. If anything, they would have been ostracized by the community for being related to the "crazy" heretic that was beaten and executed in a manner befitting "Enemy of the State". This is one scenario.
Based on the accounts of Jesus, it is very likely he would have married a woman very like minded and believing in his deity. This would have made her very devoted to him and to his mission. Their children would have been raised in the Levitical tradition in addition to all the new insights Jesus was teaching. These kids, the eldest at least, would have been in their mid to late teens by the time Jesus story climaxes with his death and ressurection.
Teens in the first century aren't like teens today. They had responsibility, understood commitment, fought in wars, married early, earned homes, worked family businesses, etc. Sure they were impulsive and passionate - hormones are hormones after all. But the point is, in all likelihood, his children would have been more like his disciples turned apostles spreading the word about who he was and aiding in Christianity's spread.
His kids wouldn't have been imbued with any special powers, just because they were his kids. But like the early apostles who were able to heal, speak other languages, and prophesy, his kids - the ones who chose that path - would have been gifted similarly to fulfill the mission they were on. And like most of the apostles in the first century they would have met a martyr's death.
In conclusion and to answer your questions succinctly...
- How would the surrounding 1st and 2nd century world treat his kids?
Depending on which way the kids went they would either be treated the same as any other kids in their positions or they would have been villified, hunted, and killed as heretics.
- Would they automatically venerate them? Some, even worship them?
Only Mary, his mother, was ever venerated. Some of Jesus' other relations did end up worshipping him and died for their beliefs. Jesus' offspring most likely would have worshipped him accordingly.
- What about those descendants that are gung-ho about spreading grandpa's Jewish offshoot religion and those that aren't?
They would have met with martyrdom in the end. Jesus line would most likely die out long before 500 AD unless one or more chose a life separate from their father's beliefs. That didn't happen very often in the 1st and 2nd century.