It is well known that the interval DI-KE in Plagal Second Mode is a major tone (12 moria). In classical written tradition (and theory), that KE is never flat. However, it is observed that in recent oral tradition sometimes KE is flattened temporarily usually to emphasise the meaning of an underlying text referring to "sorrow", "passion" or "pain". Other times it is done just for adding variety to a (repetitive) melody. It is important to note that this flattening of KE is not inherent in the mode itself, or a melodic attraction, or even a traditional temporary flattening. It is rather an innovative choice of some modern composers.

Yes. Here is an example from a Greek/Turkish song where it happens repeatedly [mp3, 4 Mb]

Yes. The most common example is in modern compositions of Leitourgika, which are well known for often being influenced by external (non-ecclesiastical) music. The following examples were taken from the Liturgy book by Archon Protopsaltis Lykourgos Petridis, a popular book with mostly modern (20th century) compositions used by many psaltai in Greece and overseas. Here is an example from the Leitourgika in Plagal Second Mode by Konstantinos Pringos (click for larger):

Another example from the Leitourgika in Plagal Second Mode by Lykourgos Petridis (click for larger)

And yet another example from the Leitourgika in Plagal Second Mode by Lykourgos Petridis, showing the flattening of KE as an optional choice in parenthesis (click for larger):

Another example is in the phrase "kai arypotois xeilesi" in the heirmos of the 9th Ode of the Canon of Great Wednesday ("Psychais Katharais"). Here it is from Harilaos Taliadoros' Holy Week book:

A. Combitsis: The example that you cite from Petridis' own Leitourgika, doesn't show up the same way in my volume. In my copy, he doesn't have that KE-flat in parenthessis above the non-flat version. This must have been a later addition to his text.