Thread: Would instrumental case be appropriate?

Native speaker has arrived!
1) Yes.
In some cases, though, we use slightly different constructions.
Студент(ученик?) страдающий дислексией
Пациент больной раком
Пассажир с большой сумкой
Заражённые СПИДом африканцы
etc.
Not all is as simple as it might seem

2) Yes,
however, afaik "syntax" in English could mean not what Russian "синтаксис" means (at least for computer programming languages). "Orthography" (орфография) might be the word you want. "По орфографии" is not correct because "орфография" is not distinct taught subject. "По биологии" would be Ok.
There is also beautiful construct "помогать в освоении английского синтаксиса" where "в освоении" is prepositional and "английского синтаксиса" is in genitive.

3) If I want to use a gerund (“Jess helped him with swimming”) should the instrumental case be used?
Swimming is just not suitable word. "Джесс помогала ему с готовкой" would be alright.

С уборкой, с озеленением, с украшением, с подготовкой, ... - all these are valid, but "c плаванием" - no.
You have to use something different:
"научила плавать", "помогла научиться плавать", "была его тренером по плаванию",...

HTH