1. So the Roman governor Felix has been looking for a bribe from Paul for a couple years. I do suspect that Paul brought a sizable amount of money to Jerusalem as a gift/peace offering to the church of Jerusalem. Paul seems to play dumb.
There are many places in the world where bribes are business as usual. Is it appropriate to pay a bribe to a government official to get medical supplies through customs and into a country, for example? It's a sign of a deficient system, yes, dare I say an inferior society. But can it simply be a cultural element that is considered appropriate for doing business in certain places, like a fee? I suspect great wisdom is necessary here, not because of abstract rules (that answer, I think, is culturally unaware of how extensively our sense of ethics is affected by our own culture) but because of long term consequences.
2. Festus forces Paul's hand. He suggests Paul go to Jerusalem to stand trial. If the earlier assassins kept their vow, they're all dead, but presumably there were others willing to take their place. The year is around AD60, six years before the Jewish War would break out and eventually spell Jerusalem's demise.
Paul's statement that he had not broken the Jewish Law or done anything against the temple or Caesar is significant. Now Luke has that on the record for posterity.
Paul appeals to Caesar, the right of a Roman citizen. He reminds me of Socrates here. Might be interesting to thumb through Plato's Apology and see if he says anything about being willing do die if he deserves it. He willingly drank the hemlock after being convicted. Acts 4 also echoed the Apology, when Peter says it is better to obey God over mortals.
3. The rest of the chapter introduces Herod Agrippa II to us. This is the son of the Agrippa who killed the apostle James, son of Zebedee. Of all the Herods, he is the only one portrayed favorably in the New Testament. Herod the Great killed the babies in Bethlehem. Herod Antipas beheaded John the Baptist.
Again, a recurring theme Luke wanted to get across to his audience, is Paul's innocent (similar to Jesus). Festus tells Agrippa that the complaints about Paul had to do with Jewish matters and his claim that Jesus rose from the dead. Agrippa wants to hear him.
4. Typical monologuing by Festus. What should I charge him with? Help me out here, Agrippa. He has done nothing deserving of death. This again is a nudge to the audience, I think. Paul should not have been put to death by Caesar. Christians are not troublemakers although others give them plenty of trouble...