It is mankind's responsibility to find solutions that are acceptable for everyone,and everything.

不是强调句
that从句是修饰solutions的定语从句，不是强调句的一部分。
翻译：
人类的责任是找到对于所有事物（everyone 译为and everything感觉还是有些欠妥）都合适的解决办法。

1st of all, mankind is not a very proper word used here, humankind is more acceptable. The sentence means that:

As a larger group, all the people have a responsibility to do something for a member of the group, or a single thing related to each and every member, until each member can accept what the larger group can do.

I hope you can get its deeper meaning, not superficially.

Good luck.

是强调句。
找到大家和一切都可以接受的解决办法是人类的责任。