Do I have puffy nipple/areola? What procedures are suitable? (photo)

Based on your photos, I would not recommend surgery to change that naturally beautiful and youthful feature that some women want but can be difficult to create by filler injection. The most I would try is Botox injection, but not sure that would do it. Areola tissue is erectile, subject to changed size and configuration in response to temperature and tactile stimulation. Peri-areolar tightening might flatten the areola but in exchange for a permanent scar of uncertain character and possible disruption of glandular elements. Aging, pregnancies, and nursing will flatten it naturally by skin expansion and subsequent involution.

Some excess breast tissue could be trimmed out from under your areolae, but you may lose sensation and this may not be worth that excision.