This question seems to be asking several things at once, but to answer to the title: yes, there exist "social" relationships where either one or all of the people involved are disinterested in sex. The term for a person who is not interested in sex is "asexual", and asexual people may be aromantic (not interested in any romantic relationship), or bi-, hetero- or homosexual (though bi-, hetero- or homoromantic would probably be a better term here). Scherrer (2008) reports on a survey of asexual people:
In this sample, eleven participants described themselves as aromantic while twenty-five described themselves as romantically oriented. Like the issue of masturbation, I did not know to include questions regarding romantic identity, however, as this data suggests, this theme emerged as important for many participants. For Mark, his romantic identity, “means [that] I separate the romance feeling from the sexual aspect.” For Mark, as well as others, there is a difference between sexual and romantic identities. Alice, a twenty-two year old white woman, describes her identity as, “Asexual. (And aromantic, i.e. no 'romance drive', no desire to find a partner).” For Alice, as well as others, claiming a romantic identity is descriptive of a person's interest in being in a partnership. [...]
While a romantic dimension might be a relatively unique axis of sexuality, asexual individuals in this survey also described their sexual identity in relation to the gender of their partner(s). Twenty three participants indicated some type of queer identity and twenty-eight indicated a heterosexual or straight gender identity. None of the self identified aromantic asexual individuals indicated gender as important in relationships, in contrast to those who identify as romantic, where all but one described the gender of their partner(s) as important to their sexual identity. Lydia, a twenty-two year old white woman, describes her sexuality saying, “I am asexual. I am also queer but that isn't about my sexuality. Just thought I'd mention that.” [...]
While only twenty-three of the ninety (25%) participants who responded to this question indicated an lgbtq identity, this nonetheless represents a relatively substantial subset of participants who describe their romantic orientation incorporating same-sex attachments as either possible or preferred. For instance, Hannah, a twenty-four year old white woman emphasized that for her, “In asexual circles, I tend to identify as asexual or asexual lesbian. In the (sexual) queer community, I tend to identify first as queer, then lesbian, then asexual lesbian.” As this description of Hannah’s highlights, while adopting an lgbtq identity is not at odds with an asexual identity, neither are these identities uncomplicatedly related. Hannah invokes the sexual identity that both fits her sense of self, and fits the community she is actively engaged with. This is not dissimilar from other findings that individuals may invoke particular aspects of one’s sexual identity as they are relevant (or less stigmatized) in a given situation Chrobot-Mason et al., 2001; (Horowitz & Newcomb, 2001; Rust, 1992; 1996).
Of the twenty-three queer individuals in this sample, eleven (48%) of them identified as bi-, bisexual, or bi-curious. This is a relatively high percentage of bi-identified individuals as in most studies that include bisexual and lesbians or gay males find much lower percentages of bi-identified people (Rodriguez-Rust, 2000; Rust, 1992). While I am not making a claim about the demographics of asexual communities, this concentration of bi-identities does illuminate the construction of asexual identities.
In addition to the article I cited, you may also want to take a look at http://www.asexuality.org/ , which has more information and perspectives on asexuality.
Kristin S. Scherrer (2008). Coming to an Asexual Identity: Negotiating Identity, Negotiating Desire. Sexualities. 2008 October 1; 11(5): 621–641. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2893352/