Q:

Do Vulcans really have their emotions in check?

How can the Vulcans claim to have their emotions in check, while many of their spieces are arrogant and condescending? Wouldn't those traits be emotionally based?

A:

I believe the answer is, some of our social rules exist purely because humans have feelings.  A society that has none (or rather is suppressing them, although that's effectively the same most of the time) would of course have different social rules.
In the case of "arrogance" and "condescension," you could say if you make a statement about yourself that is 100% correct (or at least you sincerely believe it to be) that it isn't arrogance as much as supreme self-confidence.  Even still, humans avoid explicitly saying things like, "I'm faster at running than you," because even if it's 100 percent true, emotionally, no one likes being told that they are worse at something.  But a Vulcan would never take offense at a completely true statement, as that is a completely emotional reaction.  In a society like that, there is no logical reason to be humble.
That all said, Vulcans still aren't perfect.  They can sincerely believe something that isn't true (as long as it isn't proven otherwise), have skewed values, or they can make imperfect logic, and therefore, even if they do have their emotions completely in check, they can still take actions that seem illogical to an outside observer.  The most extreme example I've seen in Star Trek is the "logic extremists" that appear in the Star Trek: Discovery episode named Lethe.  These Vulcans have such extreme values or radically different beliefs than is accepted in the mainstream, they go so far as to commit terrorist acts.  You might think this is illogical, but I believe that it's completely logical in that it's based completely on their own (flawed) logic.

