Can you play dnd with no combat? Like maybe more focused on riddles, roleplaying, puzzles, and perhaps a mystery?
Yes, if that's what you want to do the game systems can cope.
You might look into something more geared that way. Fate does pretty well, there was another one my wife was talking about that had even fewer rules than fate and would work for such a style, but I can't remember it's name at the moment.
Yes you can. You can absolutely play with no combat, focusing instead on skill challenges and puzzles.
Sure. But D&D's system is heavily slanted toward dungeon-crawls, so most other systems will be more enjoyable to do those sorts of campaigns with.