Do Tehillim follow a specific schedule corresponding to the days of the week?

Aneinu

Yes. There is a way the chapters can be divided evenly so that the entire sefer can be said over the course of a week. The order is as follows: Source: Sefer Tzeidah LaDerech, 4th maamar, klal chamishi (this source has a slightly different breakdown of which chapters end and start each day, but it is THE original source. Artscroll …

Is Tehillim written in chronological order?

Aneinu

No. The order of Tehillim is an enigma and eludes any thematic or chronological order. King David’s death is mentioned at the end of Book Two. Many lines from later chapters precede that. There are commentaries that attempt to string the flow of themes from one chapter to another, such as Rabbi Saadia Gaon.

Why is Tehillim split into five sections?

Aneinu

Chazal tells us that the 5 sections of Tehillim correspond to the five books of Moses: Bereshis (Genesis), Shemos (Exodus), Vayikra (Leviticus), Bamidbar (Numbers), and Devarim (Deuteronomy). Source: Midrash Shocher Tov, Chapter 1