How do you say "(Let us) throw the cat away, in the most polite way (습니다)?
I read that it should be '고양이를 버리십시다', but in the same exercise they changed the sentences.
Based on their second way it should actually be '고양이를 버립시다.'
I thought it was a shortening but, then they went back to 십시다.

Are they interchangeable? What is the difference? How is it formed?

What I saw vs. What I thought it should be
Which is correct?

담배를 끊읍시다
담배를 끊으십시다

앞줄에 앉읍시다
앞줄에 앉으십시다

Question

How do you say "(Let us) throw the cat away, in the most polite way (습니다)?
I read that it should be '고양이를 버리십시다', but in the same exercise they changed the sentences.  
Based on their second way it should actually be '고양이를 버립시다.'
I thought it was a shortening but, then they went back to 십시다.

Are they interchangeable? What is the difference? How is it formed?

What I saw vs. What I thought it should be
Which is correct?

담배를 끊읍시다
담배를 끊으십시다

앞줄에 앉읍시다
앞줄에 앉으십시다

Answer

"담배를 끊읍시다" and "앞줄에 앉읍시다" are correct. I haven't studied professional Korean grammar but when I hear "~하십시다", it just sounds that old people like grandparents are speaking. "~하십시다" is not sth that is more polite or formal than "~합시다" but the same. Just bear in mind that "~하십시다" is not really used these days. If I'm wrong please anyone corrects me.)

Answer

읍시다 is an ending form of Hao-che(하오체) used for inducing something(청유형).
으십시다 is an honorific form of 읍시다.

읍시다 and 으십시다 is semi-formal and can not be used for strangers or older aged person.

으시지요 can be used for strangers, especially those older or of equal age.

Answer

What is the context of this sentence?  If I were wanting to throw a cat out of the window, I do not think I would be worried about politeness...