Kefitzat Haderech: What's the Message of This Rare Form of Divine Intervention?

In parshat Vayetzei, we see an instance of kefitzat haderech, i.e., a miracle pursuant to which an individual is transported from one location to a geographically distant one with unnatural speed. Specifically, the first pasuk of the parsha (28:10) states: "Vayeitzei Yaakov miBe`eir Shava' vayeilekh Charana." On which the gemara in Chullin (91b) comments:

The Torah states, "And Yaakov left from Beer Sheva and went to Charan." And it states [next pasuk] "Vayifga BaMakom." When he [Yaakov] arrived in Charan he said [to himself], "Is it possible that I passed by the spot where my fathers have prayed and I did not pray?" When he gave his thought to return, the land contracted for him. Immediately, Vayifga BaMakom, he hit up against the place.

For a discussion of whether the kefitzat haderech occurred from Be'er Sheva to Charan, or from Charan to Beit El (where Yaakov slept) (as per the gemara in Chullin), see the parshablog.

The kefitzat haderech for Yaakov is one of four such miraculous kefitzot recorded in Tanach. A second occurred for Avraham when he was chasing the four kings who kidnapped Lot (see Rashi on 14:15).

Another occurred for Eliezer when he departed to Charan to find a wife for Yitzchak (see Rashi on 24:42).

Finally, we have an instance of kefitzat haderech in the time of Melech David when his generals waged war against the nation of Aram (see Midrash Tanchuma, Vayeitzei 3, which contends that the phrase in Tehillim 60:3 - "hirashta eretz patzamta - You made the land quake, you broke it" - refers to the divine assistance that Hashem rendered to David's generals when he caused an earthquake that shattered the land into pieces; these pieces were then brought closer together, thus substantially shortening the distance the generals had to travel to reach Aram).

What is the common denominator between these 4 instances of kefitzat haderech - and what is the message for us?

The Secret to a Happy Marriage Embedded in a Gematria

In parshat Vayetzei (28:19), Yaakov names the location of his dream - "Beit El - the house of G-d," which happens to be the future site of the Beis Hamikdash. 

In Yeshaya (2:3), the navi prophesizes that in the end of days (acharit hayamim), the nations of the world shall say, "Let us go up to the mountain of the Lord, to the house of the G-d of Yaakov."

On which the gemara in Pesachim (88a) asks, why only a reference to the "G-d of Yaakov," and not also the G-d of Avraham and Yitzchak?

A City Called Luz...

In pasuk 28:19, we learn that Yaakov gave the name "Beit El" to the location where he experienced his dream. Then, almost as an afterthought, the Torah mentions that originally this place was called "Luz." R' Hirsch notes that "Luz" means hazelnut.

What do we learn from this apparently insignificant aside?