Am I correct with the way the stack works? Since my Counterspell is the first spell off the stack, does the Standstill trigger for my opponent? Standstill says when a spell is played, and I am a bit confused by this. Is a spell still played when it is countered?
A spell is considered to be played whenever you announce it, put it on the stack and pay the costs associated with it (CompRules 409.1). Standstill will trigger at this point, when the Brainstorm is played. So, at the point you could respond, the stack consists of:
Standstill Trigger (top)
Brainstorm (bottom)
At this point, you can either choose to play the Counterspell, or let the Standstill trigger resolve. If you play Counterspell, this will cause the Standstill to trigger again. If you instead pass, then the Standstill trigger will resolve, and net you 3 cards. However, the Brainstorm will still be on the stack (CompRules 413), as only the spell or ability on the top of the stack resolves when both players pass without doing anything. Any other spells or abilities wait for the players to pass again, so you can still respond to spells or abilities on the stack, after one has been resolved.
This is a bit more technical way of answering the question, but I wanted to include the specific rules involved in the situation, to help further illustrate the situation.