For the dedication of my upcoming book, I have written
For my child.
Do you think that period should be there?
It's all based on the author, both ways work. If you like the feel of a completed sentence, punctuation is your friend! If you would rather prefer the dedication to be more like a headline, skip the periods.
sure, why not? just don't do the crazy poetry thing and hold the period until the blank page after the About the Author blurb.
More seriously, it's a purely aesthetic choice. Use it, leave it out, doesn't matter.