Nowadays we see the word “god” appear in many ways; do the same rules apply when it refers to other than God (e.g. “god of war”, “goddess”)? By rules I mean that we have to be in wudhu to touch and that we cannot throw away or burn pieces of paper on which those words are printed.

Indeed as a Muslim or a believer, God has to be always in our lives. The name must be respected and elevated. So yes, we have to be in wudhu etc