Practically speaking, one can think of it as simply breaking the contact between the foot and the surface. If one dives deeper into the milliseconds as the foot is becoming unweighted, the maximum force of friction that can be applied to our foot becomes smaller and smaller, so there's technically going to be some slipping. However, it is very minor.
When Ninjas learn the art of walking on rice paper, they're developing awareness for these tiny forces. They are learning to transfer their weight such that the back foot has very little lateral forces on it as they lift it. This sort of awareness is very useful for them.
For a more close-to-home example, we can answer your second question by considering someone walking on ice. When people slip and fall, quite often it's because their back leg slides out from beneath them. This is proof of the conjecture you had. On ice, the maximum force of friction you can attain is quite low. As people walk, they typically issue one last quick correction with their back foot before letting it rise off the ground. If this movement is too forceful, the back foot can slip.
the force of friction appears when there are two contact surfaces that interact against each other
If you by "interact against" mean "try to slide over", then correct.
As long as the threshold of this friction is not surpassed the Force of friction will adjust itself to any existing opposite force, balancing it.
Indeed. And to be accurate, every mention of friction here refers to static friction (as opposed to kinetic friction), which is what happens when sliding is prevented (when something tries to slide but doesn't).
If I understand when walking, the road will have some friction and if I exert a backwards force against the friction of the ground that same friction will apply the equal force to my foot, allowing me to move forwards.
Yes, although a bit combersome sentence. A surface does not "have friction". Rather, it has a roughness so that friction can appear when sliding against another surface wants to begin.
For this to happen, the force of friction must adjust itself to my contact force? What force does my foot exert? I'm confused...
When you apply a backwards force with your foot - we can call it a stepping force, if we will - then as per Newton's 3rd law the ground responds with an equal but opposite forwards static friction. I call it a stepping force but there is no typical official name for it as a whole, as far as I'm aware. Depending on scenario you might also call it thrust or traction or the like as mentioned in a comment.
If my "force" is bigger than the frictional force then the friction won't be able to provide that force in the opposite direction, so my foot will "continue its path" i.e. slip.
What you mean here is that there is an upper maximum limit to static friction. But note, your stepping force cannot be bigger than this limit. Your stepping force can only exist if an equal but opposite static friction force also exists (again, this is Newton's 3rd law). If the ground cannot respond with an equal static friction force then it let's go, meaning the static friction force disappears.
Then you don't have to apply any larger stepping force since your foot is just slipping and sliding and not feeling any resistance to apply force against.
But where does my force go? It makes me accelerate? I don't think "slipping" means accelerate.
As described, it doesn't "go" anywhere because you can't apply a force against nothing. When you punch into empty air then you aren't applying any force; likewise when you step backwards you can only apply a force equal to whichever resistance your feet meet.
Instead your foot just slips and slides backwards. Kinetic friction would takes over now, and then your stepping force equals that one instead. But if no other force take over - if there is no other resistance against your foot now - then you will just be "running on the spot" and will never move. As if running on ice, or if you are running while hanging in free space in a space station.
This can in no way accelerate you forwards. Only a forwards-pointing force could do that. This is the reason that it is not your step which causes you to walk forward, it is indeed the friction from the ground.
Best Answer
If there is no friction, you can still move by conservation of momentum. Take some stuff with you that you don't need. Throw it away in the opposite of the direction you want to go!