Do we find that Hashem swore about this?
Radak: His word is a Shevu'ah; He told David via Nasan ha'Navi "va'Hakimosi Es Zar'acha Acharecha Asher Yetzei mi'Me'echa" (Shmuel II, 7:12).
Why does it say "Emes Lo Yashuv Mimenu"?
Radak: This is like it says "v'Ne'eman Beisecha u'Mamlachtecha Ad Olam" (Shmuel II, 7:16).
What is the connection between choosing this place, and the promise to David's seed?
Malbim: A parable for this is a beloved of the king. Due to his love for him, the king stayed by him and built in his house a magnificent royal dwelling. He was pleased, and decided to dwell there forever. He swore never to leave the place; this entails that he will dwell by his friend's seed. The friend caused the choice of place, and the place caused the choice of the friend's seed. Also here, Hashem chose the place due to David, and due to choosing Tziyon, he swore to David that kingship will not veer from his house. There was an unconditional Shevu'ah that David's son will be king. For later generations, there was a stipulation (verse 12).
