How can a girl maintain ties with her distant relatives, and is it obligatory for her to do so? How is this accomplished, and is it sufficient to merely inquire about them through their families?

Cousins (both paternal and maternal) are not considered among the obligatory kin for whom maintaining ties is a duty, according to the preponderant opinion. Maintaining ties with them is desirable. The nature of maintaining these ties is determined by custom. Your inquiry about them is considered a form of maintaining ties if it is customarily regarded as such, while adhering to Islamic guidelines to avoid fitnah (discord/temptation). One should be keen to spread affection and avoid estrangement due due to the worldly and otherworldly benefits of maintaining kinship ties.