Is there any way to delete songs from my iPhone without connecting it to a computer?

No i don’t think so. The iphone is managed exclusively through itunes. Heres what i do if you want to delete music off the iphone: create a play list called iphone and put all the songs that i want on your iphone in that playlist. When it comes to syncing songs sync the playlist rather than your entire music library. When you don’t want a song remove the song off the playlist and resync the playlist. Hope that helps :)

Nope, everything is done through PC which is pretty annoying.

You can delete videos and podcasts without syncing but sadly, not songs.