Yes to both. There was a system in a version of D&D where, if a roll of 10 was sufficient to succeed, you could just "take" a roll of 10, without rolling. If you weren't being threatened, and you weren't under time pressure, and if the consequence of failing simply meant a lack of success, then you could take your time, starting over if necessary, until you succeeded. A thief could "Take 10" on opening a simple safe, if it wasn't trapped (or if she'd already disabled the trap), and if she had all the time in the world. If the safe was trapped, she couldn't "Take 10" to disable the trap because failing to disable the trap would spring the trap. Likewise, if a guard was patrolling and she had to get the safe open before he came back, she couldn't "Take 10" then, either.
A previous system - I think it was D&D, but it might have been another game - indicated whether a skill could be used "untrained" or not. Anybody could pick up a weapon and try to take a swing with it. But if you weren't a Wizard you couldn't try to read a spellbook no matter how smart you were. Actually, I think Rogues could gain the ability to attempt to use a spell-scroll or a magic item without really knowing what they were doing...
p.s. I feel compelled to note that you've chosen a bad example: Dancing is precisely the sort of thing you
can attempt without "proficiency." Maybe picking a lock would be a better example? A Ranger doesn't get a +2 to picking locks from her 15 Dexterity; she can't pick locks at all. But that said, yes, my whole point above was just what you wrote, that just because you have a +4 from your natural Dexterity shouldn't mean you're as good at dancing as a professional dancer, even one who is otherwise athletically unremarkable. In D&D, having an 18 Dexterity actually does mean you're a half-decent dancer, the very first time you attempt it...
That scene illustrates the difference I'm describing: The
character watches some dancers for 5 seconds and then joins them, just because she's so naturally gifted. However, the
actress playing the character has years of dance training irl.