You can power up Dragon's Crest while staying at Lv.1 by getting Zombie/Virus and fighting at Metalark Shores near Chocobo's Forest. On any Disk other than the 4th you'll encounter nothing but dragon-class enemies there, Ironites and Serpions. You can use that south gate fountain to restore Hp and Mp without losing Zombie status.

"I don't know if only Grand Dragons affect Dragon's Crest and other dragons do, as well."

All dragon-class enemies affect Dragon's Crest damage, even bosses.