1. For medical condition you can technically use c+ instrumental - Там было много людей с воспалением легких (There were many people with pneumonia there). In some cases it's better to add больные чем-либо (африканцы, больные СПИДом, but африканцы со СПИДом colloquially can be used). It's important what other words are beside this c+instrumental (the whole sentence structure is important).

2. I am helping them with English grammar - in such kind of sentence you can say Я им помогаю с английской грамматикой. But Я им помогаю по английской грамматике is also possible (the latter is mostly used when speaking about subjects , learing, etc.)