Success at all academic school subjects is highly and positively correlated. I think success in all subjects, including for example physical education, music and crafts in addition to the more academic subjects, is also correlated, but the correlation is weaker.
I am not going to write a comprehensive literature review here, but the main points are, as far as I understand (I have 60 ECTS credits in pedagogy and I am qualified as a mathematics and philosophy teacher in Finland, but I am not a professional educator or education researcher):
Socio-economic background, i.e. how rich or poor and how well educated one's family is, plays a large role. Low socio-economic status is correlated with not eating enough or healthily, with mental issues of parents and children, with alcohol use, etc. Parents of high socio-economic status tend to be more positive towards school and studies and can offer more help. There might also be more books around and reading is seen as a normal part of life.
Success in mathematics is highly correlated to one's idea of how good one is at mathematics and the way in which one things of oneself as a mathematician.
Success in reading and writing is highly correlated to how much one reads.
To the extent that "general intelligence" or IQ is a meaningful concept (there at least used to not be clear consensus about this; I am not up to date), it correlates with all academic subjects.
Self-control, being able to sit still and concentrate, etc., correlate with success in school and with motivation for school.
The following is personal speculation.
Skill at reading helps with mathematics, because mathematics uses lots of precise language, which has to be understood accurately.
Skill at mathematics helps with understanding grammar, as both deal with abstract structures and rules.
Skill with native language presumably helps with learning a foreign one.
In any case, the effect of socio-economic status and ability to concentrate on studies, and general intelligence if the concept is well-founded, support learning all academic subjects, and hence we would expect success at all academic subjects to be strongly correlated, even if they use completely different skills. Hence, we would expect someone with low proficiency in any academic subject, such as mathematics, to also do poorly on any other academic subject, such as languages. Exceptions, of course, exist.
I do not have the expertise to comment on the second question, about the necessity of exposure outside school, so I can only answer anecdotally. I learned English (from about third or fourth class) and Swedish (from about seventh class) in school. I was exposed to English via computer games, voluntary reading, later the internet, and to some extent non-dubbed movies. I was not exposed to Swedish outside the classroom. For both subjects, I did my homework and classwork. I did not really learn Swedish due to that, while my English is quite ok.