How does the Ten Commandments fit in with Paul's explanation to the Galatians?

The subject of the law and its relation to the Christian is a very important one. From what you have shared, I think you have it right. It's not about keeping the rules, but about loving God and seeking to please Him from the heart. I think we must begin by recognizing that the term "law" is used in more than one way in the New Testament.

In Romans chapter 3 (especially verses 19-22) "law" = a system by which we strive for salvation and/or sanctification through human efforts at keeping the law. Law is a system by which men foolishly strive to earn salvation.

Paul says that the law cannot save men; it can only condemn them. In Romans 7:12-14; 8:4 and 13:8-10 the "law" is God's standard of righteousness, which we are unable to achieve by our own efforts, but which is fulfilled to the degree that we walk in the Spirit. Thus, Paul loves the law as that which is holy, righteous, and good (Romans 7:12).

In the Psalms (like Psalm 119) the Law is the revelation of God's character, and His will for man. Thus, the psalmist can say, "Oh how I love Thy law" (Psalm 119:97).

Legalism is the effort to be righteous in the power of the flesh by keeping all the commandments (probably a mixture of O.T. commands, by our interpretation, and our own add-on's, the equivalent of the "traditions" of the Jews).  Grace is having a renewed heart, and thus being motivated by live and empowered by God's Spirit, and thus seeking to please God by doing His will.

The law itself is not evil, when accepted as a standard of righteousness. All of the 10 Commandments (except Sabbath-keeping) are still applicable today, as is much of the law. Those commands which are set aside are clearly indicated as such (for example the setting aside of the old food laws (Mark 7:18-19; Acts 10-11). The verses in Galatians 3 (19-29) speak of the law as God's temporary provision, until Christ came and made atonement for sins, once for all. I would suggest that you give some thought to Colossians chapter 2, as well as to the Book of Hebrews (which emphasizes the superiority of the New Covenant to the Old). 2 Corinthians chapter 3 also speaks of the superiority of the New to the Old.

If you have not read it yet, I would highly recommend John Piper's excellent book, "Desiring God: The Meditations of a Christian Hedonist." It really strikes at the heart of legalism. Amazon has it fairly reasonably: