Morph is casting a spell? How to counter it?

So, I was reading about morph and its rulins for a premodern deck with [[standstill]] and [[exalted angel]]. About the 702.37c, it got me confused. (I was reading the 708 too, but it did'nt help me.)

So, 3 little questions about morph and interactions.

If I have a Stanstill on board and morph an Exalted Angel to the board, face down, does it count as casting? It triggers the Standstill and make my opponent draw 3 cards?

Morph can be countered by [[counterspell]]? By "morph" I mean pay {3} and put a face down card. I hope I am not messing arround with the therms

Morph can be countered by [[stifle]]?

(I imagine that the first will trigger, the second, won't counter ir and the third will, but I wanna make sure I am not understending something wrong.)

Thanks!