Here are some of my thoughts on answering the questions.
1. What kinds of sermons are recorded in the New Testament? How many of them, if any, follow the pattern of our modern sermon types?
There are several "kinds" of sermons or "methods of sermonizing" recorded in the New Testament. Some might be considered topical. Several relate God's redemptive history. Some might be called testimonies. Some are or include support or defense of doctrine (Acts 15), practice (Acts 2) or are even "legal" (Acts 26). None are like the modern expository sermon. The closest might be the preaching of Philip to the eunuch. Though we are not told the details of Philip's words, we know that his teaching began with a text and a question about the meaning of that text and he expounded from there.
2. Did Jesus preach topical sermons, textual sermons, none of the above, some of the above, all or the above, or something else?
I have not fully investigated the sermons of Jesus in regard to method or style. I have a list and am going over them again. He seems to a wide variety of teaching "styles". Do a few possibly resemble the modern text-driven sermon? Possibly. Initially I'd say most all do not.
3. What kind of sermons did Peter, Paul or any of the disciples preach?
When expository preaching is considered in what is its most common definition, we are hard pressed to find Peter, Paul, Stephen, James and others -- or even Jesus -- preaching these types of sermons.
4. Does the command to "preach the word" mean any certain kind of sermon?
The command to "preach the word" cannot strictly mean text-driven expository sermons, unless we are ready to admit that Peter, Paul, Stephen and others did not "preach the word". The one who laid down the mandate in 2 Timothy 4:2 did not preach that way, as far as I find in the record.
5. Is sermon "style" driven by a scriptural mandate, personal preference, education or something else?
This question requires outside investigation or survey of preachers and is not part of textual investigations in the bible. It is my opinion (and nothing more) that text-driven expository preaching, though couched in scriptural mandate, is just as often mandated by logic, seminary training and personal preference.
6. To what extent do we follow the sermon examples of Jesus and the apostles, considering they spoke with a degree of inspiration and authority we do not have?
If the sermon examples of Jesus and the apostles are not examples for us, we have no New Testament examples to follow. Nevertheless, the imitation of these sermons must be mediated by several considerations. Our Lord and Saviour Jesus Christ is the eternal omniscient Son of God. His sermons were perfect. They are perfect examples of how a perfect God can preach and teach. He didn't just read and quote the book, but spoke with absolute authority. He said "I say unto you!" Such is too high and ultimately unattainable for fallible men preaching the word. This does not mean we do not look to Christ for guidance in preaching -- just that we must be sound and reasonable in how we approach it. The apostles spoke by direct inspiration and they also had a degree of authority we do not have. In a sense they were composing Scripture where we are reading and interpreting it. There are parts of the biblical examples we cannot attain; they are the examples we have. We preach the same message (the word), from the same authority (God), by the same power (the Spirit) to the same kinds of people (unbelievers and believers). A preacher brother explained how he sees the difference this way: one difference between the sermons we see in the Scriptures and the sermons today is that their sermons were more revelatory and ours are explanatory.
7. Were what we call sermons -- a single individual engaging in discourse or "lecture" to a crowd of people -- the basis of teaching and edification in the gathering of the church, or more for spreading the gospel of Jesus Christ to unevangelized peoples, or both, or neither?
One difference from the New Testament times is our combination of following the single preaching pastor model and the always one-sided lecture to a group of people. A single individual delivering a discourse to a crowd is seen mostly in the examples of preaching the gospel to unbelievers. But even on those occasions there is often a level of interaction between the crowd (or someone "in the crowd") and the preacher that is unseen in the modern pulpit delivery method. It is my sense that the teaching of gathered believers often took on a more interactive style of communication -- one that included dialogue and also that might be stopped if someone else had a word of teaching (Cf. 1 Corinthians 14).
I do not object to expository sermons or text-driven preaching. More of my preaching than not fits in that category. But I object to narrowly identifying this one type of sermon -- expository -- with "preaching the word". Preaching the word is presenting the truth of the word of God in many different ways. One preaching style or method does not exhaust the command to "preach the word".
