I noticed that I sometimes get dizzy when I accidentaly breathe in too much Axe. Also, I find the smell of alchohol rich deodorants quite unpleasant. But, I was wondering, is it actually dangerous? Can some chemical (alchohol) pass the blood-brain barrier and hurt my brain? Can it hurt other organs? It might be that I'm just dizzy because of the lack of oxygen...
You can definitely absorb alcohols by breathing them in. But doing do is far less efficient than drinking alcohol, so you don't have anything to worry about on that front.
To address your concern we have to answer two questions: how much alcohol is transfered per unit time and is there a local dosing effect due to proximity to the blood brain barrier.
It is probably pretty obvious that while alcohol evaporates rapidly compared to water, it still takes quite awhile. Imagine trying to inhale a glass of whiskey. You would get a serious headache but 30 minutes in you would still have an almost full glass of whiskey and no alcohol buzz. For an amusing anecdote relating how hard it is to get drunk off of inhaling even from a machine designed for it, see here
As to the second point - Fortunately, the blood brain barrier is a two way street for alcohol, which means that you can't get drunk in your brain if you don't have high levels of alcohol in your bloodstream. The two would dialyze into a reasonably balanced equilibrium.