"개구리가 설거지하고 있어."
Translation:The frog is doing the dishes.
You might be confusing a few things, let me know if I manage to hit the mark.
~하고 is a postposition that you attach to nouns to indicate connectedness. This translates to "and" or "with".
- 저는 친구하고 먹어요 = "I eat with a friend."
~고 is a postposition that you attach to verbs to indicate connectedness between verbs. This is equivalent to combining two sentences together and is common in forming more complicated forms of verbs.
- 여자는 여프고 똑똑해요 = "The woman is pretty and smart."
- 저는 공부하고 운동해어요 = "I study and exercise."
- Note that the word block 공부하고 is with by the verb stem 공부하 + 고.
~고 있다 is an auxiliary verb that you attack to verbs to transform them into a new verb that is in the progressive tense. You then conjugate the new verb.
- 개가 수영하고 있어요 = "The dog is swimming."
- 저는 친구하고 공부하고 운동하고 있어요 = "I am studying and exercising with my friend."
Note that position verbs (앉다 "to sit", 서다 "to stand", 눕다 "to lie down") have two progressive forms. The typical formation (~고 있다) is used to describe the momentary continuous action of being positioned ("in the act of sitting") while there is a special formation (Casual+있다) to describe the state of being in the position ("sitting").
- 저는 앉고 있어요 = "I am (in the process of) sitting."
- 저는 앉아 있어요 = "I am (currently) sitting."