Borchi Nafshi, Pirkei Ovos and Veyiten Lecho

Veyitn Lecho:

The saying of Veyiten Lecho is a very old minhag which is brought already in the Zohar as something said after Ve'atoh Kodosh. There are 4 main reasons brought for saying this:
1) Since this is the beginning of the week, we say Psukim of Brochos, and this is a good "siman" (similar to the "simonim" we do on Rosh Hashono which is the beginning of the year).
2) Because on Shabbos the Resho'im have a rest from Geihinom and at Motzo'ei Shabbos they go back, we say Viyhi No'am and Veyiten Lecho and in this way the Tefilla is longer, and the Resho'im have more time before they go back to Geihinom.
3) Since keeping Shabbos causes us happiness, we say these psukim that talk about Brochos and happiness at the end of Shabbos.
4) Because on Motzo'ei Shabbos we daven later then usually, these psukim are added so no one will be left alone in shul.

Minhag Ashkenaz was always to say it in Shul. The minhag of saying it at home came from the Sephardim (and was spread throughout Poland by the Chassidim).
This topic is dealt with at length in Shorshei Minhag Ashkenaz #1 chapter 8

Michael

Borchi Nafshi was said in many places in Europe (although it is not Minhag Frankfurt).
The Levush (או"ח siman תרס"ט) explains that Borchi Nafshi is said starting on Shabbos Breishis because it talks about the creation. After that the 15 Shir Hama'alos are said, because King David said them when the water from underground were going to flood the world, and with every Shir Hama'los the water went further away and back into the ground. Since the continuation of the existence of the world depended on this (if the water would not go down the world could not exist), this is also considered part of the making of the world, and that's why the Shir Hama'alos are said at this time.

Michael

The Minhag to say Pirkei ovos after mincho of Shabbos is very old, and is brought already by R' Amrom Gaon (1100 years ago), see Tur (או"ח siman רצ"ב). The minhag of saying these perokim is also brought in Rashi on Pirkei Ovos (at the beginning of Perek ו).

Michael