Here's what I think about the first two sentences. "can't help ..." means you cannot keep from doing something because of your habitual tendency, weakness, etc. It doesn't seem to fit what you're saying well. A better way to say it might be: - I feel I must be tough with them because they stopped studying. - I have no choice but to punish/scold them since they are not studying. ("take care of someone" is often used in a bad sense, so I would use it carefully.) "He didn't mind being examined yesterday: he gave up pretending being healthy." ("pretending being healthy" doesn't sound very good.) => He didn't mind being examined yesterday: he gave up (on) pretending to be healthy." => He didn't refuse/resist a physical examination yesterday: he stopped pretending to be healthy.

I’m having trouble understanding your first two examples, I’m sorry. For the last one, I would say both “Excuse me for having opened your letter by mistake” and “Excuse my having opened your letter by mistake“ are correct, although probably the most natural way to say this would be “I’m sorry, I opened your letter by mistake.”