Should it be "pepa kai", "kami kai" or something else?
I think that you are going to get a variety of answers. I think that in this case, you will actually be better using the katakana version of the English phrase 'Paper World,' which is ぺーパーワールド. Using the Japanese 界 may be confusing, as かい (kai) could be interpreted as one of many things, including large bodies of water.
Not only does the katakana form avoid potential misunderstandings (Japanese people will understand these words I think), but the English element also adds a little bit of a coolness factor that you don't get with something that is more Japanese.
Google translate "紙" as "Kami", and "Kami" as "神" when reverse. People also use "ペーパー"(Pepa) as Paper.
紙: This is the Japanese word for paper, it is read as 'kami'.
ペーパー: This is a Japanese cognate word that is used for combination words like 'Paper towel' (ペーパータオル), paper plate (ペーパープレート), and other words like this. You won't use this to describe a single sheet of paper, or a ream of paper.
神: This is also read as 'kami', but it actually the character for the word god. This will vary between religious contexts, but for the most part, it is used in the most, if not all, of the same contexts English speakers use the word 'god' (with or without capitalizing the 'G').
I digress. There are many cases where different characters have the same reading. I mentioned one case above with the reading for the world character, the sea character, and others. In this particular case, かみ (kami) could be 紙 (paper), 髪 (hair), or 神 (god). That's why Google translate does not work in reverse very well.
Things like this are also why I would never recommend that a beginning Japanese student use Google translate. Trust the textbooks, and avoid the translating tools like Google translate until you are at more of an intermediate level. If you want an online dictionary, try jisho.org
I also found "和紙" (washi), "洋紙" (youshi), and "千代田紙" (Chiyodashi). Does it mean "紙" also called "shi"?
し is another reading for 紙. You will discover as your studies progress that kanji characters have multiple readings. The reading depends on context, but as you are a beginner, I wouldn't recommend worrying about it just yet. For now, I think it is important to know that these differences exist. When you start to expand on those studies, you can start digging deeper then.