The passage is Mark 7:14-23, and Mark 7:1-13 is the event that prompted the teaching. Jews have an ancient method of ceremonial washing before they eat certain meals, which has no real sanitary value. It was part of the Jewish oral tradition, and if I'm not mistaken, it originated with Solomon. Religious Jews take oral tradition very seriously, and part of what Jesus was doing was correcting the corruption that had taken place in their oral traditions - in this case, by not washing, he set the stage for his rebuke in verses 6-13. And a little later in verses 14-23, he expanded on the heart of his teaching, and why he was doing all this. The point about food is secondary to what he said real defilement comes from - sin in your heart. That's the main thrust of what Jesus was teaching in verses 1-23 of that chapter, similar to what he was getting at on the Sermon on the Mount in Matthew 5-7.
Now that we have the context, we can look at your food question. He's talking about foods, not bodily fluids. There are some Messianic teachings among the Jews that say that when the Messiah comes, all foods will be permitted to eat again (see quotes and footnotes
here). Since Jesus is the Messiah, he began to inaugurate that. Besides, non-Jews outside of Israel are not held to the kosher standard in the Jewish law anyway.
This mainly goes to my point above - Pharisees engaged in quite a bit of "fence-building" and legalism which drew people away from the heart of what God did command and want for his people. They conflicted over this, and it's why they wanted Jesus killed. Since Paul, out of religious zeal, fiercely went after the church before he was converted, I'm sure he was no exception. After he converted, he was a different man. He seemed to honor the law privately insofar as it didn't conflict with Gospel, but always preached the Gospel first, and he even chastised Christians who were getting dangerously close to following the law to the detriment of the Gospel in their lives. Paul was also commended by Peter (2 Peter 3:15-16).
Every time Paul speaks or writes of his activities in persecuting the church in the New Testament, he does so with recognition of how deeply evil it was. This would strongly infer "yes".
No.
Yes, but see Acts 15. We must love God and do what he requires of us, and love people (Matthew 22:37-40). But God never required Gentiles to do many things in the law, so rather than do all 613 mitzvah, we have a much shorter list of what we are to do. Loving God means doing that well.