707.2a If a face-up permanent is turned face down by a spell or ability, it becomes a 2/2 face-down creature with no text, no name, no subtypes, and no mana cost. A permanent that enters the battlefield face down that wasn’t cast as a face-down spell also has these characteristics. These values are the copiable values of that object’s characteristics.
What's unclear is what happens when you turn over a DFC card. Technically you are not transforming the card (the keyword action has not been invoked), so still have a generic 2/2, but I can imagine that the Un-Rules manager might say otherwise.
The FAQAWASLFAQPAFTIDAWABIAJTBT says:
For the fourth mode, what does it mean to "turn over" a nontoken creature?
Physically turn the card (or cards, if the creature has more than one) over. If the result is a face-down card, it's a colorless 2/2 creature, the same as one would get from using the morph ability. Turning a face-down card over results in it being turned face up. Any abilities that trigger when it's turned face up will work. Turning a double-faced card over is the same as transforming it. Any abilities that trigger when you transform it will work. Turning a combined host/augment creature over will result in a big colorless 2/2 creature represented by two cards. Turning a melded creature over will result in the two cards breaking apart and forming two separate creatures, but they'll probably just get right back together even though she's too good for him. Man, we've made some weird stuff over the years. Oh, I guess turning B.F.M. (Big Furry Monster) over is the same as turning a combined creature over.
EDIT - I may have to revise that last statement. I could be wrong and technically ANY counter placed down is defined by the effect, but "put a +X/+X counter" could just be shorthand for "Put a counter on blah that increases its power and toughness both by X as long as it's there".
Not really -- if +1/+1 counters worked like you say, it would be an effect that hinges on a specific counter, and they don't do that. Counters are fungible. Even Obsidian Fireheart only cares about a presence of "a" blaze counter, not the specific one it creates.