Thank you for doing this, LearsFool. It is the reason I am on this forum; to study, discuss, and come to a greater understanding of the Scriptures. I followed your link from Acts 3 so I can start at the beginning.
My approach will be to read your questions to see what I believe I already know, then read the chapter to find and provide the answers to the questions I do not know.
1. Acts is Luke’s second book. What was his first book about?
His first book was about all that Jesus began both to do and teach, until the day in which he was taken up.
2. What convinced the apostles that Jesus had been raised from the dead?
He shewed himself alive to them after his passion by many infallible proofs, being seen of them forty days, and speaking of the things pertaining to the kingdom of God.
3. Why didn’t they begin proclaiming the gospel right away?
He commanded them that they should not depart from Jerusalem, but wait for the promise of the Father, which, saith he, ye have heard of me.
4. When would they receive power and be witnesses to Jesus?
They shall be baptized with the Holy Ghost not many days hence, but they shall receive power, after that the Holy Ghost is come upon them.
5. What would these witnesses testify about?
These witnesses would testify about the commandments that Jesus through the Holy Ghost had given unto the apostles whom he had chosen.
6. A replacement for Judas had to be named. What were the qualifications?
The qualifications to be a replacement for Judas were limited to those men which have companied with the apostles all the time that the Lord Jesus went in and out among them, beginning from the baptism of John, unto that same day that he was taken up from them.
7. How many candidates met those qualifications?
The number of candidates who met those qualifications is not given; there were about 120 gathered together. They appointed two, Joseph called Barsabas, who was surnamed Justus, and Matthias.
8. Extra credit: From what we read from this chapter, can just anyone be an apostle?
From what we read from this chapter, only those among the 120 who met the qualifications could be one of the 12 apostles. Anyone who did not meet the qualifications would be no more than successors to the apostles.