Was there a time in Jewish history when Hebrew (Lashon Hakodesh) was spoken?
Definitely until the end of the first Beis Hamikdash; consider, for example, when Chizkiyahu's representatives ask Ravshakeh to speak to them in Aramaic (the international language of diplomacy at the time), rather than in "Yehudis," the Judean language, which "the people standing on the wall" understood (Melachim 2:18:26 and Yeshayah 36:11). After that, it seems that gradually Jews started speaking other languages (notably Aramaic), though half a millennium later we still find Rebbi stating that in Eretz Yisrael, one ought to speak either Lashon Kodesh or Greek (Sotah 49b, Bava Kamma 83a), which would hardly be realistic if the former weren't still being spoken at all.
Was it only spoken by the ancient Hebrews and not the surrounding nations? If so how would said nations refer to their "non Jewish" rituals/items?
It (or closely related dialects) were indeed spoken by other nations. See Nahum's answer. That said, the way in which it was spoken, as a holy language (especially in view of Rambam's opinion, Moreh Nevuchim 3:8, that it is so called because it doesn't have specific words for bodily functions and other indelicate things), can indeed have been specific to the Jews; there's even an explanation that this is what Chazal mean when they tell us that Pharaoh couldn't learn Lashon Kodesh (when, after all, as per the Ramban quoted by Nahum, there's no reason he wouldn't have known the language of a nearby country).
If so, why don't we have non-religious literature from that period, assuming that Tanach doesn't cover the entirety of Lashon Hakodesh's vocabulary?
DoubleAA put it well in a comment: "Where would we expect to find any of that? From the secular literature club whose devotees painstakingly copied it for generations with iron age technology?" The only alternative would be a fortuitous find (similar to the Dead Sea Scrolls, and only in such a climate; in the rest of Eretz Yisrael such writings would long since have rotted away). That said, there are a number of epigraphic finds, such as the inscription on the Shiloach Tunnel, that might be considered literature of a sort. And yes, some of them do contain words not found in Tanach - for example, the Shiloach inscription has the word זדה, otherwise unknown, and suggested to mean "crack" or something similar.
Do Mishnah or other texts introduce new words in Lashon Hakodesh?
Sure. They borrow lots of words from other languages and Hebraicize them. They also create new words and new forms out of the existing Hebrew stock; the Rambam (commentary to the Mishnah, Terumos 1:1) gives as an example the fact that the Biblical Hebrew noun תרומה yielded in Mishnaic Hebrew the verb תרם (as contrasted with Biblical Hebrew הרים), which in turn is inflected like any other Hebrew verb.
Additionally, how did ancient Hebrew speakers discuss intimate actions or mundane items for which Lashon Hakodesh lacks specific words?
They might have made up (or borrowed from other languages) such words, or used the circumlocutions that Lashon Kodesh (in Tanach) does use. Or, per the Rambam in Moreh Nevuchim (cited above), there might well have been a vernacular Hebrew (unrecorded in Tanach) that would have had such words, just that Lashon Kodesh (the specifically holy register of the language) did not.
