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This question has really two parts, but they are related, so I wrote them up as a single question.
The prefix is defined in Universala vortaro (part of Fundamento de Esperanto) by Zamenhof as
ge' les deux sexes réunis; ex. patr' père ― ge'patr'o'j les parents (père et mère) | of both sexes; e. g. patr' father ― ge'patr'o'j parents | beiderlei Geschlechtes; z. B. patr' Vater ― ge'patr'o'j Eltern; mastr' Wirth ― ge'mastr'o'j Wirth und Wirthin | обоего пола, напр. patr' отецъ ― ge'patr'o'j родители; mastr' хозяинъ ― ge'mastr'o'j хозяинъ съ хозяйкой | obojej płci, np. patr' ojciec ― ge'patr'o'j rodzice; mastr' gospodarz ― ge'mastr'o'j gospodarstwo (gospodarz i gospodyni).
There is also a clarification by Zamenhof in the form of Lingva respondo 6 where he states that the prefix relates to "both sexes". He continues that while so far he has used it only of persons of the same parental line, like gepatroj and gefratoj, there is no reason for not using it of any group of persons of both sexes. Thus he gives green light for expressions like gesinjoroj.
Since the definition and the clarification speak about "both sexes" and by the time of Zamenhof there was just two recognised sexes, the established meaning of the prefix has become "at least two persons with at least one male and one female".
The world has changed from the time of Zamenhof and some consider that there should be a way to express these changes of how we regard gender and sex. There are a lot of reform proposals for gender marking (I counted at some point eleven more or less thought-out proposals): some introduce small changes, some big, some create a parallel system of naming, some alter/reinterpret the meaning of current names for relatives.
One of the most widely spread proposals is just allowing singular ge- indicating a person of whatever gender. As Neil Roberts points out this could cause confusion. There is a longer argument in PMEG against singular ge-. Personally I think the problems are exaggerated.
Noteworthy is also that the need for ge- is shrinking. At the time of Zamenhof the sentence Mi renkontiĝos kun miaj amikoj meant meeting only male friends, nowadays many take it to mean meeting whatever gendered friends. Quite likely ge- will remain in use of the closest relatives only, while some for instance already use kuzo of any cousin, not just of a male one.
The affixes ge- and -in are incompatible, because what would a word with markings for "both sexes" (plural ge-) or "any gender" (singular ge-) and "female gender" (-in) mean.
The ge- prefix modifies the word so that it refers to “both” sexes. This is most often because the following word part is plural and describes a group of men and women, like gefratoj. However, it doesn’t have to be plural, for example geedzeco is a common word for marriage. It is using ge because edzo is one of the few gendered words in Esperanto, so when describing the state of a pair of people of different genders being married, it is useful to add the ge- to distinguish it from just the state of being a male husband.
Some people do try to use a word like gefrato to describe a sibling in a gender-neutral way, but this is adding a new meaning to the ge- prefix and it isn’t traditional Esperanto. It can cause confusion with words like geedzeco which would traditionally be understood to be involving “both” genders. With the new meaning of ge- the word becomes ambiguous because it could mean the word is about the state of being a married person without specifying the gender. Sadly there isn’t yet a universally accepted single word to express a sibling in a gender neutral way. Some people have proposed neologisms like sibo for this. Take a look at parentismo for more information.
I can’t think of any examples of where using ge- with -in- would make sense. Perhaps if you accept the new meaning of ge- mentioned above, then gepatrino could be the same as patrino, but then the ge- is redundant because you are making the word neutral with the prefix and then un-neutralising it again with the suffix. If you use parentismo instead then it could make sense to say parentino if you want to adopt a style of speech where you avoid using the gendered roots altogether but want to specify the gender in this particular case.
I have seen now and then many uses of the singular and personally I don't see any objection. Logically if "la gefratoj" means "the siblings" (both the brothers and sisters) and "gefratoj" means "siblings" (brothers and/or sisters) as is the established usage (the presence of one sex or another is allowed as an obligatory possibility and not an obligatory actual presence), I see no objection to "la gefrato" meaning the sibling with equal possibility allowed for him to be of one sex or another or even refuse to identify with one as a non-binary person or more commonly as an examinee wanting his identity to remain anonymous even sex-wise.
There is a second meaning of ge meaning the presence of both members of a couple of exactly two, giving "ge" a more sexual connotation. But even then I would allow the singular to means either sex. La geedzo would mean a spouse without presuming whether it is the husband or the wife.
