3. Is the concept of "senior pastor" biblical? If so, where do we find it? I agree that there are leaders singled out among the elders, but is this one leader?
In your first question, it depends on what you mean by 'biblical.' Do we find the title itself in scripture? No, not that I'm aware. Do we find some patterns in Scripture or is the function a necessary entailment of some teaching in Scripture? I would say "yes."
First, as you point out, the Scripture itself makes the distinction between all elders and those elders whose job is preaching and teaching (1 Tim. 5:18). "Double honor" is "especially" owed to those whose job is preaching and teaching. So, it would seem that there is to be at least one elder who plays that role in a local church. There could be more, but there must be at least one. That person would be the "primary preaching elder" or what we popularly call the "senior pastor." But if there is more than one who serves well in the word as this text presumes, then is there any grounds for establishing one "senior pastor"?
Second, there are the historical patterns in the NT itself. The most frequently cited (and abused) example of this is Jesus' closer relationship with Peter, James and John, with Peter serving as the most frequent leader and spokesperson of the twelve). Alexander Strauch sees at least two other examples in the situation with the deacons in Acts 6 and in the relationship between Paul and Barnabas (whom Strauch regards as an apostle equal to Paul). So, there does seem to be some historical patter of "first among equals" in the NT itself.
Third, there are cases where a local church only has one qualified teaching elder for a given period of time. That's the case with both Timothy and Titus as the ministries are being developed in Ephesus and Crete. Until the eldership is really established and trained, and in God's kindness other teaching elders are raised up, Titus stands as the sole elder who must "straighten out what was left unfinished and appoint elders in every town" (Titus 1:5). That could take years; in which case, you would have one senior pastor or teaching elder doing all of the teaching. And even after elders are appointed, there would be grounds for that man to continue as the main teaching elder since he'll be training the others.
Fourth, with the churches in Revelation, the Lord addressed the "angel" or "messenger" of those churches (Rev. 2-3). We know that Ephesus had a plurality of elders in place since the time of the apostle Paul. But the address goes to a single messenger rather than to the elders as a group. Apparently one main shepherd is to communicate the message of God to the people of God.
Fifth, Paul addresses his strongest charge to "preach the word" to a single elder, Timothy (2 Tim. 4:1-2). Anticipating a context where false teachers will abound and the people will want teachers to tell them what they want to hear, Paul calls on this one man to be faithful in the midst of all of that. This is a pragmatic inference, but I think there is much to commend a primary preaching elder when it comes to warding off error and focusing the people on sound teaching.
It's interesting--and I don't know what to finally make of it--but it seems to me that most often when the NT addresses elders as a whole (Acts 20:25-31; 1 Pet. 5:1-4, for example) the main instruction is not a call for all of them to teach (again, 1 Pet. 5:17 assumes that won't be the case) but for them all to shepherd or oversee or care for the flock. This is an inference, but when I compare the direct addresses to a single elder to teach (Rev. 2-3; 2 Tim. 4) with the addresses to the elders as a group to shepherd or oversee (Acts 20; 1 Pet. 5), there does seem to be a working assumption that there is one elder with primary teaching responsibility in the congregation.
4. What do you think of Alexander Strauch's (Biblical Eldership) position on this?
Well, I'm no Alexander Strauch. He's a better scholar and writer than I would ever be. While I'm mentioning him, let me plug his book Leading with Love. An excellent meditation on 1 Cor. 13 and Christian leadership.
In brief, as I understand it, Strauch considers Jesus' relationship with the three (and Peter's leadership among them), Acts 6 and the prominence of Phillip and Stephen among the seven, and Paul's relationship to Barnabas as arguments for primus inter pares (for a free on-line booklet see here). He calls for a primus inter pares based not on official office but based upon a kind of prominence that comes from superior giftedness and dynamism when compared to the others. There are two weaknesses to this view in my opinion.
First, none of the texts in view actually say that "first among equals" status is established according to superior giftedness. It's an inference that Strauch draws in consideration of the unusual giftedness and events involving these individuals. The text is silent on this matter.
Second, if we take this functional basis for primus inter pares we're inevitably left with a debatable subjective criteria for establishing this role. We have no consistent way of determining leadership of the leaders, or "first among equals." For example, in the case of Peter, Peter arguably is not the best leader given his quick tongue and nature. John, perhaps, would be more loving and therefore a "better" subjective choice.
What I do like about Strauch's approach is that it reminds us to not pigeon-hole any one elder as "the" leader on every issue. We should pay attention to gifting and unique suitedness for particular leadership roles and tasks as the elders jointly carry out the leadership of the church. Put your best man forward for the particular issue at hand. That's wise. And it keeps the church from relying too much on one man or imposing unreasonable demands on one man. We should keep this in mind, even if we would suggest a different basis for "first among equals."