Who Can Baptize?

Last Sunday the church I'm attending had a baptism service. In one of the baptisms, a father baptized his son. That got me thinking:
- is it biblically allowable for a non-church leader (e.g. pastor) to perform a baptism?
- should baptisms be connected to a local church?
- should baptism be preceded by a period of discipleship/testing/training or immediately after salvation?

Then I read Kevin DeYoung's post on the first question above. I greatly appreciate DeYoung's blog and the courage and insight he portrays on it. This is not a "gospel" issue, but a fun one to think about and discuss. So I thought I'd weigh in on his four arguments:

1) He says: "Biblically, we see that those who perform Christian baptism in the New Testament have been set apart by Christ for an office in the church (e.g., Peter, Paul, Phillip). Strictly speaking, the Great Commission, with its command to baptize, was given to the apostles, not to every believer indiscriminately. There is no evidence to show that private members baptized."

First, the most natural reading of Acts 9:18 is that Ananias baptized Paul. We have zero evidence that Ananias was an elder.
Second, Phillip was not an elder. Some may consider him a deacon, though that specific title is not used in Acts 6. In fact, the only two men from Acts 6 that are described after that passage are described as preaching the gospel: Stephen and Phillip. Philip seems to be more of an evangelist. He may have been a proto-deacon, but that is different from being an elder.
Third, the logic seems to be that every one explicitly described as baptizing in Scripture is a church leader, so only church leaders should baptize. If that's the case, would DeYoung use the same logic to say "since only adult believers are explicitly described in Scripture as being baptized, only adult believers should be baptized"? I'm doubting that ...
Fourth, while DeYoung has clarified his comments about the Great Commission being given "to the apostles," I still find his thoughts on this wanting. If we are following "good instincts" by "obeying" the Great Commission by sending missionaries throughout the world, then couldn't we also be involved in baptism more directly? Regardless, I do believe the Great Commission was initially intended for more than just the apostles. I believe every believer should be involved in going, making disciples, baptizing, and teaching. That doesn't mean that every believer will go into the jungle and start a church, but on some level they are to be involved in this process. Nothing in the Great Commission seems to limit the teaching or baptizing to only the apostles or the "church leadership".
Finally, D. A. Carson says the following regarding the Great Commission in his Matthew commentary (page 666): “The injunction is given at least to the Eleven, but to the Eleven in their own role as disciples (28:16). Therefore, they are paradigms for all disciples. Plausibly, the command is given to a larger gathering of disciples. … Either way, it is binding on all Jesus' disciples to make others what they themselves are—disciples of Jesus Christ.”