The correct answer is $\boxed{\text{A}}$.
The dot product of two vectors $\mathbf{u}$ and $\mathbf{v}$ is defined as $$\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \theta,$$ where $\|\mathbf{u}\|$ and $\|\mathbf{v}\|$ are the magnitudes of $\mathbf{u}$ and $\mathbf{v}$, respectively, and $\theta$ is the angle between $\mathbf{u}$ and $\mathbf{v}$.
Statement 1 is correct. The dot product of two vectors is always less than or equal to the product of their magnitudes. This is because the cosine of the angle between two vectors is always less than or equal to 1, which is the case when the vectors are parallel.
Statement 2 is incorrect. When two vectors are perpendicular to each other, then their dot product is zero. This is because the cosine of a right angle is 0.
Statement 3 is correct. The dot product of two vectors is positive or negative depending whether the angle between the vectors is less than or greater than $\frac{\pi}{2}$. This is because the cosine of an angle is positive when the angle is less than $\frac{\pi}{2}$ and negative when the angle is greater than $\frac{\pi}{2}$.
Statement 4 is incorrect. The dot product is not equal to the product of one vector and the projection of the vector on the first one. The projection of a vector $\mathbf{v}$ onto a vector $\mathbf{u}$ is given by $$\mathbf{v}_\text{proj} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|^2} \mathbf{u}.$$ The dot product, on the other hand, is given by $$\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \theta.$$ Therefore, the dot product and the projection of a vector onto another vector are not the same.