I am not an expert, but I will just leave here what I have found.
For 겸양법 (Some people call it "object honorification," but I am not sure whether it is a proper translation), Middle Korean had these three characters that were attached to verb stems when the endings started with a consonant (ㅂ became ㅸ when the endings started with a vowel):
ᄉᆞᆸ (when the stems end with ㄱ, ㅂ, ㅅ, ㅀ, or ㅎ)
ᅀᆞᆸ (when the stems end with a vowel, or ㄴ, ㄹ, or ㅁ)
ᄌᆞᆸ (when the stems end with ㄷ, ㄵ, ㅈ, or ㅊ)
According to 백문식 who wrote an etymology dictionary, those pre-endings (-ᄉᆞᆸ-, etc.) came from the verb 다. meant 白 (to mention, to state, etc.).
At that time, the pre-ending -ᅌᅵ- was used for 공손법 (They call it "addressee honorification"). For example, 먹ᄂᆞ니ᅌᅵ다 is now interpreted as 먹습니다. Some characters including ㅿ, ᅌ, and ㆍ disappeared (ㅿ and ᅌ disappeared earlier than ㆍ) and honorification changed over time, which affected the characters I have just mentioned and the following happened:
-ᄉᆞᆸᄂᆞᅌᅵ다 ＞ -ᄉᆞᆸᄂᆡ다 ＞ -ᄉᆞᆸ늬다 ＞ -습니다
but, in consideration of several entries and example sentences in a dictionary, I can supplement the history with the following:
-ᄉᆞᆸ- + -ᄂᆞ이다 → -ᄉᆞᆸᄂᆞ이다/ᄉᆞᆸ노이다 ＞＞ -습니다 (used after a consonant but not after ㄹ, ㅆ, or ㅄ) (1933) ＞ -습니다 (used after a consonant but not after ㄹ) (since 1989)
-습니다 is used for the "addressee honorification" not for the "object honorification." This shows a change in honorification.
The history of -습니까 can be said as the following:
-ᄉᆞᆸ- + -닛가 ＞ -삽닛가/습닛가 (1934) ＞ -습니까 (since 1957)
-ᄂᆞ니잇가/ᄂᆞ닛가 (after a stem) and -니잇가/닛가 (after a stem or a pre-ending) were used to form "yes or no" questions. -습니까 can be used with or without interrogatives. It is quite natural that ㅅㄱ or ㅺ became ㄲ; for example, 아까 was once written as 앗가.
As for -읍시오, some believe that, after -ㅂ시다/읍시다 had been formed, -다 of -읍시다 got replaced with -오 to form imperative sentences.
-(ᄋᆞᆸ)사ᅌᅵ다 ＞ -(ᄋᆞᆸ)새이다 ＞ -ㅂ새다/ㅂ세다 ＞ -ᄇ시다 ＞ -읍시다
-읍시- + -오 → -읍시오
Nonetheless, it does not explain why -습시오 does not exist (-ᄉᆞᆸ시- existed before).
김현주 claims that people who were not well-versed in the court language used -ᄋᆞᆸ시- for the "addressee honorification" when the subject of a sentence was equal to the addressee. It seems that the author believes the following:
-ᄉᆞᆸ- + -으시- → -ᄋᆞᆸ시-
According to the dissertation, such an unordinary combination became possible when -ᄉᆞᆸ- became used for the "addressee honorification" because -으시- was (and is) used for 존경법 ("subject honorification"). If this is the case, the absence of -습시오 is quite understandable.