Yes, provided you can mass produce sufficiently long carbon nanotubes or similar; but the power cable going down that space elevator is also about the length needed to encircle the earth if it was on the ground, so if you can build that part alone you don't need the tower.
This isn't a "no": if you're building a space elevator anyway you probably do want to put solar on the top. It's just you don't want to build one only for the solar on the top.
This isn't a "no": if you're building a space elevator anyway you probably do want to put solar on the top. It's just you don't want to build one only for the solar on the top.