the "month" to grow up is a loose average, it may take a bit longer, but only by a few days!
And yes! you will just log in and find yourself as an adult! hope this helped ^^

As Tigerart said, sometimes it's not exactly the "30 days" we associate with "one month". The game actually calculates "a month" by taking the date you joined and changing the month in front of it. For example, if you joined July 4th, your deer will grow on August 4th, even though the 3rd would mark 30 days (since July has 31 days in it). It's also calculated based on the first time you log into the game, not the time you register or enter the community site.

Dancing, I've noticed that you have some art of Ysrael as your icon. This piece by her player (useless-machine on deviantART) was designed for her own use, and really should not be used without her direct permission, as that is considered art theft, which is taken very seriously in this community (many of us are artists!). You can read more about the matter here.
Thank you for understanding!