Is there a name in Spanish for the inner part of the elbow (on the opposite side of the part we call "elbow")? If not, how would it best be described?
Yes, there's a name (although not very commom): sangradura o sangría. Here are the (relevant part of the) definitions according to the DRAE:
Although those are the "official" names, they are rarely used and it's usual to refer to this part (at least in Colombia) as el pliegue del codo.