Of course it's not a blunder, it's not even a mistake, perhaps an inaccuracy. THe knight had to move somewhere as it was under attack my two white pieces. I suppose c6 could have made the knight stand, but it would have been a prime target of attack, losing multiple tempi most likely in its defensive process, taking saves tempo, while allowing white to recapture towards the center for some control later on, in exchange for more rapid development for black and no major weaknesses out of the opening.

Opening theory-wise it's a blunder. Nb6 is normal.

Qa4 Nc6 Nd7 or Bd7 all run into Qxe4+ so yes the pawn is immune but there was a better resource than the lines you posted. Nf3 doesn't solve the e4 weakness problem and d3 allows for the dsb of white to come out on a nice diagonal although perhaps it belonged on b2 or a3 after a4?

white is focusing too hard on castling?

A common amateur mistake: move pawns in front of your king without castling first. My plan involved e4 and d4, which requires my the king to be safely away and the king side rook to support the center advance. In closed games it often pays to delay castling, but this is going to become pretty open in a few moves.