Yes, you can take 고 as that when 있다 follows the sentence. However it's not used with an adjective verb. 덥고 있습니다 doesn't make a lot of sense while 더워지고 있습니다 does make perfect sense, 'it's getting hot'. -지다 is a passive-verb form similar to 'get adj.', not exactly the same.

yes, you're right. the word "고" can mean to show "~ing" in English but i think it will not make any sense if you think of "고" as a seperate form. usually when it come to showing "~ing" in Korean, it always comes with "~하고 있다". "~하고 있다" is a typical form of "ing" and means "somebody is in the middle of something." Also we can say "~하고 있습니다" as a formal one of "~하고있다". let me give you some more examples of "~하고 있다" or "~하고 있습니다." 전 지금 당신의 질문에 대답하고 있습니다. i am answering your question now. 대답(noun) + 하다 (it is a one way to make nouns become a verb adding "하다" on nouns) 대답 하고 있다. 대답 하고 있습니다.(more formal and polite) let's see pure verbs. 전 지금 TV 을 보고 있습니다. i am watching TV now. 보다(watch) 보고 있다(be verb + watching) 보고 있습니다. 읽다(read) 읽고 있다 읽고 있습니다. now we've got two types of verbs which is "noun+ 하다" and "pure verbs" and it's easy to make "ing form" of them. in case of "noun +하다"', it goes "noun+하고 있다" as taking out "다" as in "하다" and adding "고 있다" in pure verbs the same rule can be applied. :) hope it could be helpful.