The waves are nothing but a physical vibration of air molecules. So when an object moves through the air, it interacts with the air molecules like how a boat or a submarine interacts with water.
In your case of a moving object, these air molecules get compressed in the direction in which the object moves. This is because the object has a velocity and this keeps pushing the air molecules as it moves. And like you said, the frequency of the waves will be higher at the front of the object. But at the opposite, the frequency of the waves will be lower as they are not packed up together (imagine the same boat and wave analogy). Yep! This is the doppler effect.
And in your second case of a stationary omnidirectional sound source and a stationary observer, the frequency of the sound will be constant in all the directions. Why? Because the object is stationary and it’s not compressing the sound waves in any direction. The sound waves have their original frequency. The volume at the observer’s end will be less audible as the amplitude of the sound wave is inversely proportional to the square of the distance – aka inverse square law.
And here is an interesting case! When the object starts travelling closer to the physical maximum limit of the molecules could travel (speed of sound), they keep stacking up together like a barrier. And when the object travels faster than this limit, it breaks that barrier and causes a sonic boom. Read more on how sonic boom works.
