How does the torah reading and haftorah reading relate for Va-yayshev? Why do we read from Amos after reading from Va-yayshev?
Levush (Orach Chaim 669) says that it is because its first verse condemns the Jewish people על מכרם בכסף צדיק - for selling the righteous for money. This ties in, then, with the brothers' sale of Yosef (even though the simple meaning of the verse is referring to the kingdom of Yisrael some 900 years later).
Indeed, the next phrase, ואביון בעבור נעלים, "and a poor person [they sell] for shoes," is also seen (in Yalkut Shimoni, Bereishis 142) as hinting at what the brothers did with the money they received from the sale - they bought shoes. (Which, in turn, is the source for the Midrashic account of the Roman emperor "filling his palace with shoes" and then calling the Ten Martyrs to task for that crime.)
The Rav Pinches Friedman Chlit"a did a very interesting drasha on this question.
See shvileipinches for more information.
The Targum yerouchalmi says that Yossef's brothers bought shoes with the money of the sell. Why do they choose specifically this object?
The Rav Chlit"a quotes the Middrash Rabba in Wayiqra which says that Women were protected from Arayoth in Egypt by Sarah imenu's zechout (she preserved herself from Arayoth with Paro). Men were protected by Yossef's zechout (Potifar's wife...). Yossef is like a shield for Bene Israel regarding the Arayoth like shoes are a shield regarding the land. The first sentence of the Haftara talks about 3 poshim and not 4. The 600000 neshamos in Egypt are gilgulim of neshamos from Soddom veAmora who are themselves gilgulim of Dor hapallaga (Babel tower) which are themselves gilgulim of Dor Hamaboul. The prophet says that with the zechout of Yossef, those neshamos won't need an other gilgul. The complete drasha is available in English