A little late, these were started on Sunday...
4:1 So that, my brothers, beloved and desired, my joy and crown, be standing in this way in the Lord.
This verse could either serve as a conclusion to what has gone before or the beginning of what follows. Since what immediately follows does not seem to spell out how to stand in a general way, we think it goes more naturally with what precedes than with what follows. The particular conjunction (hoste) also seems more often to continue what precedes than to initiate new trains of thought.
Accordingly, the sense would be Paul telling the Philippians to "stand" or to live as he has just instructed, following his example, having their citizenship in heaven, waiting for the redemption of our bodies. The Philippians are his joy because they are, for the most part, models of what it means to be a believer. They are his crown perhaps in the sense that they demonstrate the success of his ministry, for which he might in one sense receive honor in the great athletic contest of faith.
4:2 I urge Euodia and Syntyche to think the same in the Lord.
We do not know anything about these two women except what is said in this and the next verse. They were Paul's "coworkers" at Philippi in some way. The passage knows nothing about fixed roles for such women. Paul simply says that they worked with him as other men did.
We have seen the idea of "thinking the same" before in Philippians. Paul urged the Philippians in 2:2 to think the same and to be of one soul and mind. He then went on to give them the example of Christ, who although he had the status of God was willing to take on the role of a servant, even to the point of death.
4:3 Yes, I ask also you, genuine yokefellow, join in helping them, who worked together with me in the gospel, also with Clement and my other coworkers, whose names [are] in the book of life.
Paul's failure to give the name of the person in question is intriguing, leading some to suggestion that the word "yokefellow" itself is actually a person's name, Syzygos. However, what a happy coincidence that would be, that a person's name fit so nicely with his function! Since we are not persuaded that Christians received new names when they were baptized at this point, it seems more likely that Paul is referring to a specific person.
However, there is one other person from the church at Philippi who has been named thus far in the letter, the person who very likely delivered Philippians back to Philippi. This is of course Epaphroditus (2:25). Therefore, it seems more likely than not that Paul is addressing Epaphroditus here. Paul is urging Epaphroditus to help Euodia and Syntyche to get along.
We have no way of knowing who the Clement mentioned in this verse might be. It seems quite doubtful that it is the same Clement who would later become bishop of Rome at the end of the century.
This is the only time in Paul's writings, indeed the only time outside the book of Revelation, that the book of life is mentioned. In Revelation 17:8, this book includes the names of those who would be saved, known and at least figuratively written down in God's mind since the creation (cf. Dan. 12:1). The obscure reference is a reminder that much that is not mentioned in any New Testament writing, including the entirety of the Pauline corpus, is at least possibly assumed. At the same time, we have no secure basis to know what those things were.