with regards to the bolded part, no (or actually, it depends). Note that units will decrease some stats on AW-ing, but some abilities like +7% archer attack (Spica) and increase def buff (Deine) may offset it.
like buttlover and kotono said, there are 2 Awakenings. I prefer to call them Unit Awakening and Skill Awakening.
Unit Awakening = unlock/upgrade unit ability + decrease some stats but increasing max stats (AW20~30 to reach 50cc60 stats I believe)
Skill Awakening = upgraded skill. you can ignore this portion entirely for now, we will not expect to get it within a year from now.