Q3. (Acts 13:1-3) Why are the church leaders in Antioch seeking God through corporate prayer and fasting?
Corporate Prayer and fasting honors the ordinance of Christ in Matt 18:19-20 "Again I say unto you, That if two of you shall agree on earth as touching any thing that they shall ask, it shall be done for them of my Father which is in heaven. For where two or three are gathered in my name, there am I in the midst of them." This has been a regular manner before and after the day of Pentecost (See Act 1:13-14; 2:1, 42-47; 4:23-31).
How does God respond to their seeking?
God spoke to them and revealed His will to set apart Paul and Barnabas for the work He has for them, to carry the gospel to the gentiles.
Why do you think we don't do more of this sort of thing today?
We rely on our own effort rather than that of the Holy Ghost. Human planning has supersede God's own.
What was the importance of this to the church at large? To the chosen missionaries?
The importance of seeking God is to help us serve God in the best manner and to have His stamp on all our undertakings for they would then be under His guidance. As a result of this, we won't be doing things by human wisdom but by the demonstration of the Spirit and the power of Holy Ghost. This will bring much fruit and success to the mission works.