I have been wondering if there exist gender differences in German like female and male dog, as in Spanish there is a clear difference between el perro (male) and la perra (female).
Yes, there are names for different genders of animals.
1. One method is to add the female ending "-in":
der Löwe (the lion), die Löwin (the lioness)
der Hund (the dog), die Hündin (the female dog)
der Hase (the rabbit), die Häsin (the female rabbit)
2. There are several animals which have special names for males and females
I'll give some examples here in the order: general name of animal, the male, the female, the child (cub, kitten etc.)
der Hund (the dog), der Rüde (the male dog), die Hündin (the female dog), der Welpe (the puppy)
die Katze (the cat), der Kater, die Kätzin, das Kätzchen
das Pferd (the horse), der Hengst, die Stute, das Fohlen
das Rind (the cattle), der Stier/Bulle, die Kuh, das Kalb
das Huhn (the chicken), der Hahn, die Henne, das Küken
das Schwein (the pig), der Eber, die Sau, das Ferkel
das Wildschwein (the boar), der Keiler, die Bache, der Frischling
You can find more examples here: https://de.wikipedia.org/wiki/Liste_der_Bezeichnungen_f%C3%BCr_Haus-_und_Wildtiere