If we go into more detail, 하고 already includes a verb, as it is 하다(do) + -고(verb ending for "and" connection). In other words, apart from the particle -하고(and, with), 하고 as a verb connective form is also very widely used. You can think of phrases like 숙제 먼저 하고 티브이 봐 (Finish your homework first and then watch TV). 하고 here is just a verb conjugation and has nothing to do with the particle 하고. Your example should be said as 문을 열고 닫았어요, as -고 by itself fulfills the "and" connection role. But if you mean "opened and then closed (sequence)", there's even a better way: 문을 열었다(가) 닫았어요. 열었다(가) 당았다 is clear on "open first, then close", while 열고 닫았다 may mean either enumeration or sequence. Also, 하다 is an interesting verb in that it can stand in for many other verbs (말하다, 얘기하다, 되다, etc). Even the particle -하고 can be thought of as an evolution of the verb 하다. That is, 너하고 나 (you and me) = 너(를) 먼저 하고 나 = 너를 먼저 넣고(include) (그 다음에) 나 (counting you in first and then me).