Just like the trope, the tunes are traditional, and yes, each community has its own tunes and even its own system of chanting. A chazan (cantor) also has significant leeway in choosing tunes. Since the text is set but the tune is not, the chazan can simply compose a new tune or adapt someone else's tune. There are halachot regarding this, but not everyone observes them. Generally, there will be a specific mode (called nusach for Ashkenazim) for chanting prose text, similar to reading from the Tanach, and poetic text may be set to an actual melody.
The way Ashkenazic nusach works is that there are some very basic melodic motifs that the chanting should generally use, and the chazan is free to improvise within that framework. As synagogue services become more lay-led and less often led by professional chazanim, these motifs solidify into melodic snippets that essentially get rearranged. There are lots and lots and lots of examples at offtonic.com/nusach, including melodic snippets, freer mode-based improvisation, and composed melodies.
The Sephardic world is vast when it comes to diversity, and different communities have very different practices. Spanish and Portuguese communities do essentially the same as Ashkenazic communities; there are some melodic snippets that the chazan rearranges to fit prose text, and there are composed melodies for poetic texts. Some tunes actually straddle that line; the famous tune for Bendigamos, for example, gets quite stretched over the course of the song! Moroccan communities tend to add much more embellishment and improvisation while essentially keeping to the same format. Syrian communities have a much freer approach; there is a specific musical scale (the maqam) used for each service, which depends on the service, the week of the year, etc., and the text is chanted by improvising freely within that scale. The improvisation tends to follow certain characteristic patterns, but they don't even come close to being melodic snippets. For poetic texts, they sing some composed melody in the same maqam. You can listen to examples at Sephardic Hazzanut. Yemenites... Yemenites do their own thing. I can't pretend to understand Yemenite chanting.
To answer one of your other questions, generally speaking Biblical quotes (which are all over the liturgy for what should be obvious reasons) are not specially marked by melodies. There can be exceptions. At one Yom Kippur service I once attended, the sha"tz (stands for SHaliach TZibur, the prayer leader) decided to sing the Hashivenu line of Sh'ma Kolenu in Eichah trope. I've also heard, at Rosh Hashanah services, I heard the musaf paragraphs detailing the day's sacrifices chanted as it was done during maftir, in high holiday Torah trope. In many Sephardic communities, including Moroccan as well as Spanish and Portuguese, the Kabbalat Shabbat psalms are chanted with T'hilim trope, whether they know it or not! Also on Friday night, Moroccans (and only they, that I know of) read four lines from Song of Songs right after L'chah Dodi, and those four lines are read with Song of Songs trope.
I hope that answers some of your questions! NusachDB is a pretty great resource for melodies from around the world. I know the guy who runs it; I sleep next to his wife every night! (: NusachDB has a subcategory Nusach Resources, which links to over 400 sites (I don't remember my last count, but you can check) with recordings. You can even find some sheet music (other than my transcriptions, I mean), including the famous German compendium of nusach from the 1870's, Abraham Baer's Baal T'fillah. These chants and melodies are always a combination of the very old and very traditional, the very new and very traditional, and the simply new. I hope you can learn some of these melodies; Judaism, globally, is far the richer for having them.