If a body in a vacuum is at rest in a flat surface (velocity is $0$), then it experiences no force due to friction. If the body has a velocity then it does experience a force due to friction. This indicates that the force due to friction is a function of velocity. What is this relationship?
Is it also due to acceleration? Higher order derivatives of position?
Understanding the relationship between the drag, which is the force of friction in the context of fluids, and the speed of relative motion is simpler than understanding the relationship in the context of friction between the surfaces of everyday solid objects. The law of static friction referred to in the question is at best a semi-empirical law. Let us try to understand this statement by repeating a thought experiment in combination with empirical knowledge about this phenomenon, which is elegantly described in the lecture on the webpage linked in this answer.
If we take a block with known weight $W$ and place if on an incline, we can change the angle of inclination $\theta$ fairly easily. At a particular inclination, we observe that the block slides down the incline with an approximately uniform speed. Using the Newton's first and second laws describing forces and our knowledge of trigonometry, we can then define the coefficient of friction as $\mu := \frac{W \sin{\theta}}{W \cos{\theta}} = \tan{\theta}$.
Clearly, executing a real-world experiment which behaves in this ideal fashion is exceedingly difficult, but we can observe this behavior in some time intervals of the block's motion over the incline. However, even if we are able to execute such delicate experimentation, it does not completely describe the underlying phenomenon. For instance, consider a metal block made of a single element (as pure as possible) placed on an incline constructed identically in a vacuum, with both contacting surfaces having identical roughness. In this case, we may observe that the block and incline fuse together. In effect, our ansatz for the law can describe semi-empirically, the behavior of a variety of surfaces with impurities and varying roughness which are in direct contact, but is incapable of precise predictions for general surfaces. Similarly, verifying that the coefficient of friction is independent of the relative speed between the surfaces, is a challenging task which requires delicate experimentation and is not a completely answered question at present.
The answer to your question is: It depends.
There are different kind of frictions
These do not depend on the velocity. So, taking your first example: If a body of mass $m$ rests on a flat surface in vacuum, you need to overcome the adhesion before the body starts to move. The friction is usually accounted for by $$ F = \mu F_n = \mu \, m g $$ where $g$ is the gravitational acceleration, and $F_n = m g $ is the force normal to the surface.
There exists other types of friction which depend on velocity. However, they are usually accounted for, if one considers fluid dynamics. Hence, in your example (body in vacuum) the friction is (to a good approximation) independent of the velocity.
