This means he was careful to find out exactly what happened. He probably talked to the different people who saw what happened to make sure that what he wrote down about these events was correct. Another way to translate this would be "carefully researched everything that happened"
Luke said this to show honor and respect for Theophilus. This manner of address may also mean that Theophilus was an important government official. Other ways to translate this would be "honorable" or "noble" Some people may prefer to also put this greeting at the beginning and say "To…..Theophilus" or "Dear… Theophilus"
This name means "friend of God." It may describe this man's character or it may have been his actual name. Most translations have it as a name. (See: [[rc://en/ta/man/translate/translate-names]])