You seem to be asking for a definitive verse in scripture that states or implies that the soul has a gender. There is no such verse (it seems), and indexes to mentions of gender list nothing concrete. However:
God made humans, "in the image of God he created them; male and female he created them" (Genesis 1:26), and this was so before the fall. This passage is used to show that human persons are male or female.
The body will rise, for "It is sown a natural body; it is raised a spiritual body". This passage is used to show that the perfection of a person includes both soul and glorified body. A person is not soul, not body, but the unity of both.
A soul is a male soul in the sense of being the soul of a male person: a person with a male nature. Your hand has no gender, your blood has no gender. In that sense a soul has no gender.
- The Bible's account of creation assigns gender to persons. Human persons are both body and spirit. So in that sense gender is not purely physical, because humans are not purely physical.
- Souls have "types" in the sense of being the souls of female and male humans, of animals (non-eternal gendered souls), of angels (no gender), of God (no gender!, though the human nature of Christ was male (...but this merits its own question), and we have been instructed by Christ to refer to God the Father as "our Father").
- Masculinity and femininity can be aspects of souls/spirit. That the perfections of the female as well as the male can be found in the perfection that is God, who is spirit and without gender, can be seen in Isa 49:14-15; 66:13; Ps 131:2-3; Hos 11:1-4; Jer 3:4-19 (CCC270). Which perfections are appropriate to male humans and which to female might merit its own question.
- Are there masculine and feminine souls? Of course: the souls of human males, the souls of human females.
- For similar reasons, gender is eternal. A person who has her breasts cut off, like St. Agatha, does not cease to be a female. Even a person who has his head cut off, like St. John, does not cease to be a male. Adam and Eve will rise still male and female, as they were even before sin.