Can I perform system commands within my PHP code on these servers? For example: $temp = exec("./executable"). If the answer to the previous question is yes then, would a ./ be required to execute a program like in the previous example or no? In other words, would I have permission to run system commands? I am curious because I like to write some programs in C++ that I use in conjunction with various websites.

Yes you can do it.

Best regards,
Slavik Koshelev