Your maximum hand size is checked only during the cleanup step of your turn. At any other time, you may have any number of cards in hand.
Countering a spell that has been cast will not prevent you from drawing the card.
If multiple effects modify your hand size, apply them in timestamp order. For example, if you put Spellbook (an artifact that says you have no maximum hand size) onto the battlefield and then put Recycle onto the battlefield, your maximum hand size will be two. However, if those permanents entered the battlefield in the opposite order, you would have no maximum hand size.
To play a card means to play a land or to cast a spell that's a card (and not a copy of a card).