When the AD curve shifts to the right, it means that people are willing and able to buy things at a higher price level. If people are willing and able to buy things at higher prices, then of course prices will go up because businesses will rise their prices (all other things being equal).
So imagine that people get a bunch more money (in aggregate). They are more willing to buy stuff and so prices go up. If supply does not go up to keep up with this demand, prices will rise -- this is demand pull inflation.
Yes, price level in the economy rises when aggregate demand curve shifts rightwards.
The shifting of aggregate demand curve to the right implies that the aggregate quantity of a product demanded by customers at any given price has increased as compared to the previous levels. So when the demand curve shifts to the right, while the supply curve remains unchanged, the market equilibrium point represented by the point at which the demand curve intersects the supply curve shifts to right side on the supply curve. As the supply curve is an upward rising curve, this means that both market equilibrium prices and quantities increase.