Personally, I would be surprised if this isn't a topic elaborated upon in the DMG. In the meantime, however, here's what I've been doing with my players:
First of all, the whole debate comes down to the question "Can I intuit anything about spells another individual is casting?" As a DM, I've been allowing players to use an Intelligence (Arcana) check to determine certain properties about a spell. Restrictions to the check are that the player in question must aware of the casting and not incapacitated. Furthermore, any loss of senses gives the player disadvantage on the roll. I do allow the check to be a part of the reaction/casting process for Counterspell The DC is not fixed for the check, but instead based on thresholds as follows:
Result of 10 or better - Player learns the spell's school
Result of 15 or better - Player learns the spell's base level
Result of 20 or better - Player learns the exact spell name
Result of 25 or better - Player learns the exact level at which the spell is being cast
I would also like to point out that nowhere in any D&D text does it indicate that somatic or verbal spell components are the same from class to class, (or species to species for that matter) even for the same spell. So a wizard's fireball might look and sound completely different from the same spell coming from a Light-domain cleric. Thus, I hesitate to give any advantage to the Intelligence (Arcana) check based on class. Instead, I explain it to my players as though the Counterspell-caster is somehow interpreting the flow of magical energies.
With regard to your last point, about trying to determine the efficacy of Counterspell in combat, as a player I personally only cast it at Lv 3. Augmenting the level of Counterspell confers no advantage on the ability check to manually counter the spell, and I've found that the Lucky Feat gives me a great edge in that regard. That being said, I would consider changing this strategy once I begin encountering Lv8+ spells regularly, as those are less likely to be countered manually even with a re-roll.