Can BTT trigger upon text selection like popclip?

no, but it is a good idea for a new trigger. It is complicated and requires many heuristics though because there is no system notification for when text selection has changed.

Does Popclip only work when selecting text with the mouse or also when selecting via keyboard? (I assume they listen to mouse drag events and then use some heuristics to determine whether text was selected. That's probably also why it doesn't work with all apps (Troubleshooting — PopClip))