If someone asks me how something is or how I am doing, for example if someone asks me "wie geht's?" Is it okay to respond with "alles gut" or should I say "Alles ist gut?"
Doesn't really matter, there are several options and as those are short, daily phrases, no one expects you to form out lengthy sentences. However, it is common to thank and to ask how the other person is doing:
Danke, gut, und Ihnen? (more formal)
Danke, gut, und selbst? (less formal)
Just observe what other people say, as there are regional customs.