The problem with this question is that static friction and kinetic friction are not fundamental forces in any way-- they're purely phenomenological names used to explain observed behavior. "Static friction" is a term we use to describe the observed fact that it usually takes more force to set an object into motion than it takes to keep it moving once you've got it started.
So, with that in mind, ask yourself how you could measure the relative sizes of static and kinetic friction. If the coefficient of static friction is greater than the coefficient of kinetic friction, this is an easy thing to do: once you overcome the static friction, the frictional force drops. So, you pull on an object with a force sensor, and measure the maximum force required before it gets moving, then once it's in motion, the frictional force decreases, and you measure how much force you need to apply to maintain a constant velocity.
What would it mean to have kinetic friction be greater than static friction? Well, it would mean that the force required to keep an object in motion would be greater than the force required to start it in motion. Which would require the force to go up at the instant the object started moving. But that doesn't make any sense, experimentally-- what you would see in that case is just that the force would increase up to the level required to keep the object in motion, as if the coefficients of static and kinetic friction were exactly equal.
So, common sense tells us that the coefficient of static friction can never be less than the coefficient of kinetic friction. Having greater kinetic than static friction just doesn't make any sense in terms of the phenomena being described.
(As an aside, the static/kinetic coefficient model is actually pretty lousy. It works as a way to set up problems forcing students to deal with the vector nature of forces, and allows some simple qualitative explanations of observed phenomena, but if you have ever tried to devise a lab doing quantitative measurements of friction, it's a mess.)
To determine whether friction is static or kinetic, examine whether the two surfaces are moving relative to each other -- if so, the friction is kinetic, and if not, it's static. It doesn't matter which inertial reference frame is chosen. Even though the ground is an inertial frame, the relative motion between the box and ground means that friction is kinetic here.
This makes sense in the context of the physical difference between kinetic and static friction. Static friction usually results from two surfaces "interlocking" together into a configuration that requires additional force to change. On the other hand, kinetic friction usually results from chemical interactions between the two surfaces. The interlocking can only occur between surfaces that aren't in relative motion.
Best Answer
The static and kinetic friction are phenomenological forces that have the same microscopic origin, viz., the EM interactions between the molecules.
The "static" friction is still there, just that it's magnitude is reduced, and so given another name.