@kuwaly's answer is essentially correct.
I'd just like to add that I've never heard of the 7th floor being excluded from a building. The major numerical superstitions in Japanese (and probably Chinese, etc.) surround the numbers 4 (四 yon or shi) and 9 (九 ku or kyuu), since they sound like "death" (死, shi) and "suffering" (苦 ku). To the best of my knowledge, there are no similar superstitions surrounding the number 7 (despite the fact that, as you correctly point out, 7 can be read shichi).
Also, just to make this clear - the presence of "alternate" readings like yon and nana (for 4 and 7 respectively) isn't purely because of superstition. Rather, it's because there are two parallel schemes for counting in Japanese, one of which uses native Japanese words (hito, futa, mi, yo, itsu, ...), and one of which uses Chinese imports (itsu / ichi, ni / ji, san, shi, go, ...). This is vaguely similar to how English has native words (one, two, three, four, five...) and Greek/Latin imports (mono / uni, di / bi, tri, tetra / quad, pent / quint, ...). If you're interested in learning more, consider asking a question on Japanese.SE