It doesn't matter if you use adjectives or not in this game. The noun has a gender. The gender of the adjective just agrees with the gender of the name.
Nouns in Spanish can have two genders, masculine and feminine. From your examples,
corazón is masculine ("el corazón) and
cara is feminine (la cara) (Note that
sonriente is an adjective that has the same form for both genders).
Masculine gender is not just used to refer to masculine individuals, but to refer to the class of the individuals. For example, if I say "los alumnos" I might be referring to both sexes, not just male students. If I say "las alumnas" I'm exclusively referring to females.
Realize also that there are nouns that are common for both genders ("pianista", "dentista"). You could use any gender for these ("el/la pianista").
Again, from your examples, you would use "el oso" or "los osos" to refer to bears. If you had a picture of "mother bear", where it is evident that is a female you can use "la osa", but otherwise masculine usually refers to the class. There are exceptions, of course ("las abejas", "las hormigas"), and only few nouns are ambiguous about gender ("el/la mar", "el/la editorial").
So, I would say that there is no special way of handling the nouns, and it should not be difficult to tell which gender to use which for the description of the flashcard or which card to get given a description.