What is the significance of the 'Voice that they heard in the Garden? What was Hashem doing there?
Seforno, Rashi and Targum Yonasan 1 : Hashem: Kevayachol, was walking round the Garden, as He did every day, seeing to its needs, and that is what Adam and Chavah heard and went to hide.
Ramban: Whenever the term "Mis'halech" 2 (the reflexive) is used in connection with Hashem, it refers to a revelation (or the departure) of the Shechinah, accompanied by a mighty wind. 3 Here too, the Shechinah was accompanied by a strong wind (See 3:8:2:2 and 3:8:2:1).
This seems to tally with a list of commentaries cited by the Ramban who learn like this - in fact, it was the voice that they heard walking round the garden.
Rosh: Mis'halech is like Mes Holech - the angels said that Adam is like a Mes. Hashem answered "l'Ru'ach ha'Yom" - Arvi'ach (I will widen) his day to live to be like Mine (1000 years).
The wind here was not a mighty one, so as not to frighten and confuse Adam and Chavah.
What is the meaning of the words "le'Ru'ach ha'Yom:
Rashi: It refers to the west direction, where the sun was about to set at the end of the day.
Ramban: It refers to the strong wind that accompanied the Shechinah (refer to 3:8:2:2)
Why did Adam and Chavah hide when they heard the Voice of Hashem in the Garden?
Seforno: In fulfillment of the Pasuk (in Ki Seitzei) "And He (Hashem) shall not see in you any nakedness".
Targum Yonasan: Because they were actually trying to hide from Hashem out of shame (as is evident from his explanation of Pasuk 10). 1
Since they now had a Yeitzer ha'Ra, whose strategy is to make a person believe that he can hide from Hashem.
The Gemara (Chulin 139b) asks "what is the source in Torah for Haman", and answers "ha'Min ha'Etz." How is this connected to him?
Da'as Zekenim : Hashem said that it is proper to hang Adam on the tree - rather, He will hide it, and later hang Haman on it.
רש"י: שהיה מתהלך בגן: למה הוסיף רש"י מילת 'שהיה'?
גור אריה: על קול נכון לומר 'הולך' [במשקל 'פועל'] ולא 'מתהלך' כי בנין מתפעל שייך רק בהולך מעצמו ולרצונו, לכן כתב רש"י שמילת 'מתהלך' מתייחסת אל הקב"ה 'שהיה' מתהלך בגן. 1
בנין 'מתפעל' הוא פועל יוצא אל עצמו- מה שאדם עושה לעצמו ולא מה שאחרים עושים בו, ולכן זה לא מתייחס אל הקול.
רש"י: לאותו רוח שהשמש באה משם: למה זה נקרא "רוח היום"?
גור אריה: השמש נקראת יום כי היא העושה את ה'יום', ובאותה שעה הייתה השמש במערב.
כיצד ניתן להתחבא מהשכינה?