I used to see music theory as overly complicated jargon that recapitulates the same concepts in seemingly disparate forms. While that's not completely untrue, any thorough discipline will have overlapping vocabulary. Modes are definitely one of those things... somewhere in the intersection of scales, keys, octaves, intervals and harmonics, modes pop up. I tend to try to reduce processes to their most basic forms. Hopefully this dissection will add to your understanding of how modes can be used.
A mode is a pattern of intervals starting at any particular root note (the key). A section of a mode that repeats each octave is a scale. Scales overlap... a lot. Every scale can be expressed as a transposition of another scale in a different mode. A motif is a recognizable "unit" of "thematic identity"... typically a short rhythm or clever syncopation stretched over a few note intervals. Each note in a motif is in the same key & mode... and when the motif is repeated and transposed to a different key, the mode is preserved in that new key. Usually, some artistic transitions are used to soften the repetitiveness of the motif... also preserving mode. These repetitions, transpositions, and transitions form a melody.
The transpositions mentioned above do not necessarily need to be in the same key & mode... i.e. a melody can change key. That's a progression. A melody can certainly change modes too, though that can get intense quickly. I don't believe in deciding to write key progressions before melodies, or vice versa... an understanding like this enables many creative approaches to composition. It is common for musicians to be able to compose a "one-note" melody - as this is simply a natural vocalization of a basic mental process (humming or singing a song in your head). Composing this way often leads to simple sounding music because the resulting melody is either in one key, or transposes so frequently the composer cannot identify supporting harmonies without a thorough understanding of music theory and so supports only the key & mode of the root note (or safe things like 5ths & 3rds).
Also, a composition does not need to stay in the same mode. Often a mode change can be employed to separate parts of a song (e.g. verse from refrain).
As far as genre specifics go... no, there's nothing typical. With music "at large," some modes are more prevalent than others, especially in certain cultures... but I don't think this has anything to do with their appropriateness for a particular genre. From what I understand, modes add a type of musical organization the brain recognizes as cohesive... and that's as far as it goes. Because a motif in one key/mode may overlap completely with another key/mode, how is one to tell them apart? It's often the inclusion of one defining interval that gives a melody is identity as being "in" a mode. It's really only when all of a mode's intervals are traversed like a scale (or at least played in close proximity) that the "cultural sound" of a mode becomes apparent... which is to say, an adept composer can "sneak in" the use of modes (and benefit from more complicated, aurally aesthetic musical structures) without overtly coloring the sound of their composition. Mat Zo, Dyro, Naten, Feed Me... doin' this stuff all the time... every time.
