Rashi writes that Avraham, not knowing his guests were angels, asked them to wash their feet before entering his home lest they be pagans who worshipped this dust and would bring their idolatry into his home. It could be that the Torah (or more accurately, Rashi) simply means to underscore the degree to which Avraham distanced himself from idolatry, but (as my son’s Rebbe asked yesterday) why mention specifically the detail of “worshipping dust of the feet”?
He goes on to give a very "pnimiyus" explanation of this, which I'll admit is not really up my alley. I certainly don't think that the answer given reflects Rashi's intent, but rather, that the question serves as a pretext to introduce some interesting kabbalistically oriented divrei Torah, about "katnus hamochin".
I would attempt to answer the question, on the level of peshat, in a different, straightforward way. On a peshat level, why does the Torah relate that Avraham said that they would be able to wash their feet, when the pasuk states:
|ד יֻקַּח-נָא מְעַט-מַיִם, וְרַחֲצוּ רַגְלֵיכֶם; וְהִשָּׁעֲנוּ, תַּחַת הָעֵץ.||4 Let now a little water be fetched, and wash your feet, and recline yourselves under the tree.|
I would say that it is part of hospitality, rather than as part of Avraham's avoidance of avodah zarah. Indeed, this is not my own innovation. Besides being the obvious peshat of the pasuk, Rashi says as much:
|and bathe your feet: He thought that they were Arabs, who prostrate themselves to the dust of their feet, and he was strict not to allow any idolatry into his house. But Lot, who was not strict, mentioned lodging before washing, as it is said (below 19:2): “and lodge and bathe your feet.” - [from Gen. Rabbah 54:4]||ורחצו רגליכם: כסבור שהם ערביים שמשתחוים לאבק רגליהם והקפיד שלא להכניס עבודה זרה לביתו. אבל לוט שלא הקפיד, הקדים לינה לרחיצה, שנאמר (יט ב) ולינו ורחצו רגליכם:|
Thus, Lot was not so strict about it. So why did he even do it? Because of hospitality. The derasha that the concern was idolatry, in that they worshiped that dust, was made on the basis of the switching of the order, between the incident with Avraham and the immediately following parallel incident with Lot. (See the midrash inside.)
And indeed, in perek 19, we see Rashi state:
|and stay overnight and wash your feet: Now is it customary for people to first stay overnight and afterwards to wash? Moreover, Abraham said to them first, “and wash your feet!” But so did Lot say (i.e., he reasoned), “If, when the people of Sodom come, they will see that they have already washed their feet, they will invent false accusations against me and say, ‘Two or three days have already passed since they came to your house, and you did not let us know!’” Therefore, he said, “It is better that they remain here with the dust on their feet, so that they should appear as though they had just arrived now.” Therefore he said, “Stay overnight” first and afterwards, “wash.” - [from Gen. Rabbah 50:4]||ולינו ורחצו רגליכם: וכי דרכן של בני אדם ללון תחלה ואחר כך לרחוץ, ועוד שהרי אברהם אמר להם תחלה (יח ד) ורחצו רגליכם. אלא כך אמר לוט אם כשיבאו אנשי סדום ויראו שכבר רחצו רגליהם, יעלילו עלי ויאמרו כבר עברו שני ימים או שלשה שבאו לביתך ולא הודעתנו, לפיכך אמר מוטב שיתעכבו כאן באבק רגליהם שיהיו נראין כמו שבאו עכשיו, לפיכך אמר לינו תחלה ואחר כך רחצו:|
Now it is possible that Rashi didn't write the above comment. (See this parshablog post, the update at the bottom of that post, and the comments at the bottom of that post, for elaboration.) However, if he wrote it, the assumption therein is that there is a natural order of doing things when one has guests. And that is that it is customary to first wash and only then stay overnight. So this is basic hospitality.
Why then, does Rashi discuss idolatry? On the basis of a midrash, which is in turn on the basis of the switching of the order between the incidents by Avraham and by Lot.
Why mention the specific detail of "worshiping the dust of the feet?" Because Rashi, and the midrash, worked with what the Biblical text gave them. The verse already mentioned the washing of the feet, for other peshat reasons. How, then, could Rashi or the midrash be expected to mention a different form of idolatry? And it was because he suspected them of being Arabs who did this worship, and thus he wanted to eliminate this particular idolatry which was present. But that does not mean that there is some deep meaning to the type of idolatry mentioned!
I think this answer is so good that it undermines the very question. Of course, there is a different I like to draw between prooftext and pretext, and if I were asking the question, I would be regarding it as pretext. Regardless, it was a great question to ask, because it helps us get to a truer understanding of the peshat and just what it is that Rashi, and the pasuk, are saying.
On the other hand, the "danger" in posing questions such as this is that where the true answer to the question can be found by challenging faulty assumptions, but no one realizes this, and instead the answer given is the one matching the pretext, one can become trained to treat trivial questions as insurmountable, the faulty assumptions as axiomatic, and the "deep" and contrived answers as compulsory, as the text otherwise makes no sense.