"To think to oneself" is a set expression, both in Russian and in English. In Russian it would be думать про себя or, indeed, just "думать". After all, you rarely say "SOMEBODY though to himself, full stop". Normally you add what it is they thought to themselves. And in such situation you may choose to fall back to just "think" because this is an action that is automaticlly presumed to be happening only in one's head.
сам suggests what it says on the tin: the person thinks on his own, without using people's help or their insistent and maybe annoying advice. Or, well, he might behefit from their advice but prefers to do his own thinking first. That's what сам essentially means: performing action on one's own, yourself. Depending on the context, English might express it as "on one's own", "oneself", "by oneself" and even "for oneself" (=trying something out to see if it is as people say).
Точно подмечено )) Added an alternative pronunciation here: https://forvo.com/phrase/%D0%BC%D0%BE%D0%B9_%D0%B1%D1%80%D0%B0%D1%82_%D0%B4%D1%83%D0%BC%D0%B0%D0%B5%D1%82_%D1%81%D0%B0%D0%BC/
Here is a sentence that demonstrates the difference between "think" and "believe" in English. "My brother believes for himself." doesn't work. But "I think the children want water." and "I believe the children want water." are the same thing. Do you mean to tell me that I think the children want water, but don't believe it? I think not!
Й and И are two separate letters in the Russian alphabet. It's not an accent mark. And the meanings are different. Мой брат is my one brother. Мои братья are my brothers (plural). The possessive pronoun has to match the singular or plural noun that they modify. You used the plural "my[мои]" for a singular noun "брат." Мой is the singular.
It's not about not caring about others, it's about not relying on them in making decisions. As in "he forms his own opinion and doesn't blindly follow any leader".
"He cares for himself only" would be "он думает только о себе".
In general adding "сам" to a verb means that an action is performed without guidance or assistance.