Is this used in the sense of a child choosing a partner for school, or can can it be applied to a relationship status as well?
I probably do not quite get the meaning of "partner for school" (is it a mate sharing the same bench in shool? or a tutor who helps in studies?) - if so, none of this exists in Polish.
"Partner" (male noun) means:
A person with whom one does something together (dances, plays a game, has a conversation, starts an enterprise, drinks alcohol).
An actor who is playing on a stage as a supporting actor to another one.
A person with whom one has a relationship (busines, sexual, etc.): "On jest moim partnerem biznesowym" [He is my business partner (contractor)]; "On jest moim życiowym pertnerem" [He is my life partner] - but this is a very formal expression, so in everyday situations it is rather avoided.
Someone who has a status, competences or powers equal to someone else.
Attention: "partnerka" (female noun) is used only in the first 3 meanings. In the 4-th meaning male noun is used even if related to a woman: "Anna (female name) okazała się twardym (male adjective) partnerem (male noun) w negocjacjach' [Anna proved to be a tough partner in the negotiations]. Due to this, in the business relationship (3-rd meaning), a male noun "partner" can be also applied to a woman to stress her equality in the partnership.
Attention2: Expression "Być dla kogoś partnerem" [to be a partner for somebody] indicates directly the 4-th meaning. In negation, it has a specific nuance: one who "is not a parther" has lower competences, so: "On nie jest dla mnie partnerem" [He is not a parther for me] means that he is weaker than me, while "Je nie jestem dla niego partnerem" [I am not a partner for him] means that I am weaker than he.
'Have you a partner?' which I entered and which was rejected, is good English.
"Partner/partnera" is also used nowadays in Polish as person who you are in a more intimate relationship. It's pretty much equivalent with a fiancee or a boyfriend, so it should accept partner or boyfriend/fiancee, as it can be either one of those.
Well yes, it can be a romantic partner, absolutely true. However, the word itself is totally vague. That's why we want it to remain vague in English. After all, saying "partner" doesn't really say much about the level of intimacy.