Do I know what spell is being cast?
Not through the counter spell itself. In general if there is a specific ability to do X D&D 5e specifies it through keywords or description. See Specific beats general on Page 7 of the PHB or Page 4 of the D&D 5e Basic rules.
Do I at least know (somehow intuit, based on VSM components) the level of the spell?
The Arcana skill is your best candidate. Unlike spells, feats, and class abilities, the specifics of when and where the various D&D skills are applied are left to the Dungeon Master to decide. On page 175 of the PHB and is 74 of the D&D 5e Basic rules state that the every tasks a character or monster may attempt is covered by one of the six abilities. The following section lists suggested specifics but it is not meant to be taken as a definitive list of everything a character can attempt in game.
The assumed default is the player to describe what they are doing, and the referee comes up with the rulings to resolve it. Sometime this means saying no if it the action is impossible given the circumstance or the amount of time.
Or can I tell the level of the spell by sensing a disturbance in The Weave? (I.e., if the spell is being cast using Subtle Spell, can I still tell what's going on?)
This is considered flavor text and is left completely in the Dungeon Masters to describe how the various ability checks manifest in his campaign. Some would opt for a mundane description, other would do as you did above and use various bits of background to spice up their description.
To make a ruling you need to consider is how long the identification takes. If you consider it like a search then it would require a character spend his action on his turn to figure out what was cast. If you consider it to be quicker then you may be comfortable in letting a spell identification check occur during the reaction.
You can use the box labeled Improvising an Action on page 193 of the PHB or Page 72 of the Basic rules for guidance in deciding how to handle the check.
My personal opinion is that I would allow a check but the character would have a very high DC of 25 to ID the spell during a reaction. If they get a 20 or higher they get one piece of information during the reaction. Otherwise they just have to cast the counterspell knowing only that a spell is being cast. If done as part of their normal turn I would drop it down to DCs of 15 and 20.
Another suggestion I have is that you could opt to fully describe the material component being used and allow player skill to come into play. I would do this anyway has I find the various description of material components evocative.