When translating "l?r?a (l'ruach)? hayy?m" (hayom, Genesis 3:8) as "in the breath of the day", this seems to suggest to me the birth of the day, as the day starts to come to life, before the sun rise. If this is correct, would this be a good time to devote to seeking HaShem?

It's usually translated as the direction of the sun in the afternoon, which in fact is said to be an especially good time to pray from Isaac (Breishit 24, 63) and Elijah: (Kings ii,18, 27) and the entire story there. One explanation is that one stops his work day in the middle (especially in winter) to relate to God, others mention the majestic feeling one has at a beautiful sunset! Nevertheless, we should face the land of Israel which is (usually) in the east.