When a character with a come into play effect (like Focused Art Student) enters play insane, does the come into play effect still takes place?

So when i excaust Things in the ground and i draw a Focused Art Student which enters play insane, may i draw one card?

I think I may because the insanity and blanking of the cards text takes place after it is put into play. The "enters play" effect triggers as soon as it leaves my hand and is put into play. Am I right?

Unfortunately no, you are slightly mistaken. The "put into play" bypasses all other effects (except uniqueness). So the character will enter play insane and no "enters play" effects occur. Similarly, willpower and terror icons offer no protection from entering play insane.