If I wrote a textbook, or other form of academic material, is it okay to use problems found online?
In general, no. You may not republish materials found on the web unless you attend to copyright laws in your jurisdiction. These laws vary quite a lot, but for your intended purpose - inclusion in a book - you will need permission in most places. 

Sometimes the original publisher will explicitly give permission on a web page, such as a Creative Commons permissive license or a statement the the material is public domain. But even a license comes with restrictions, such as, perhaps, attribution. 

Sometimes the original publisher will include something like terms of service like these from Lamar University. Your use is bound by those unless you make other arrangements with the copyright holder. 

In the absence of any statement to the contrary, you should probably assume that the creator or other copyright holder "reserves all rights". 

When in doubt you can and should contact the creator of any materials. Both of the links you provide actually have contact information. 

In addition to copyright law, for which you can be sued, there is the question of plagiarism. It is considered unethical to claim any work created by others as if it were your own. 

I'll note for the record that there are a few exceptions to copyright law, but they don't include republishing the work of others. If those problems have "value", then that value is owed to the creators. 

Also, copyrights generally expire, but the terms are normally very long - something close to 100 years at this point. 



There is one subtlety, however, that partly explains why you see the "same" questions in multiple places. Note that it is words and "expression" that is the subject of copyright, not ideas. The only "ideas" that can be protected from reuse are those covered by explicit patent. That is a completely different thing. There are some things, however, for which there may be only one way to properly express it in a given language. 

"Integrate f(x) = sin(x)" 

is one of those things. The sin function and integration are ideas with common terms. There are very few other ways to write that problem. So it will appear in that form or one closely similar to it in most calculus books. In general, the ideas are free to use. 



Finally, you seem to be asking for legal advice and no one here can give it. In particular, no one here can give you permission to do something that is illegal to do under civil or criminal law. I am not a lawyer. My advice is conservative to help you avoid stepping into a problem that is best avoided.