What is the exact meaning of 0.7V voltage drop in a forward-biased diode?
when the diode is forward biased, holes are pushed from P to N and electrons from N to P, correct?

Yes. [However, &quot;forward biased&quot;, really only means that the P side is more positive than the N side. When the forward bias is very small, &quot;leakage&quot; current may predominate. Leakage current is caused by minority carriers. Although majority carriers are (usually) more common (hence &quot;majority&quot;) they have a harder time crossing a PN junction than minority carriers. So, when the bias is very low, the few minority carriers that easily cross the junction may form a larger current than the many carriers that have a hard time crossing the junction.]

what do the charge carriers injected in each semiconductor do after they have passed the discontinuity?

They become minority carriers. There are multiple fates for such minority carriers. They may recombine with majority carriers, they may diffuse back across the junction, they may survive and reach the metal-semiconductor junction.

Do they recombine and produce heat?

Whatever their fate, they will eventually interact with the crystal lattice, with other carriers, with metal, etc, and their trajectory will become randomized. That is they will produce heat. However, the electric field will cause those carriers or new carriers to accelerate, thus maintaining the current.

How can the KCL law be true if some carriers disappear at the discontinuity?

When an electron and a hole recombine (at the discontinuity or elsewhere) there is electron current in one direction and conventional current in the other. So algebraicly, current is conserved.