To put an enemy under someone's foot means to conquer them. Alternate translation: "Yahweh was helping David to defeat his enemies" or "David was busy since Yahweh was giving him victory over his enemies" (See: [[rc://en/ta/man/translate/figs-idiom]])
The word "name" is a metonym for the person, and "for the name" refers to worshiping the person.See how you translated this in [1 Kings 3:2](../03/02.md). Alternate translation: "in which people would worship Yahweh" (See: [[rc://en/ta/man/translate/figs-metonymy]])
This speaks of Yahweh giving David complete control over his enemies as putting them under his feet. Alternate translation: "Yahweh was enabling David to completely defeat his enemies" (See: [[rc://en/ta/man/translate/figs-metaphor]])