Do Japanese (pop) songs usually preserve pitch accent (so that the words have to be chosen in accordance with the melody, just the words in English songs have to match the meter) or does it get completely overwritten by the melody? Or do both pitches interact in some way?
"Singing", as in what people might do with their voices in songs, can vary a lot by genre, style, etc. When people say words in different ways, contrasts in speech, such duration between long and short vowels (e.g. o vs ou) may become harder, if not impossible, to distinguish. As mentioned in some of the comments above, pitch often gets overridden by musical concerns (pitch of melody, and so forth) in many songs.
It should be noted however, however, that something like "pitch accent" is more than just a change in pitch. Pitch accent might be accompanied by all sorts of things, such as voice quality, duration, etc. (I don't know the specifics of this for Japanese pitch). While "pitch accent", or some other term, might be a convenient way to talk about the structure of Japanese in theory, these contrasts are natural phenomena which are signaled in many ways (not just some +/-HIGH, as a grammar text may describe it).
In the end, for people proficient in a language, recognizing words even with added musical "noise" may not be a problem. More so than with Japanese, I've seen this with Chinese. A foreigner would not get very far learning the distinguishing tones for words from some song, but a native speaker or experienced learner would have no issue repeating lyrics in a normal speaking voice, with the "correct", expected tones.
The simple answer? No. Japanese completely overwrite pitches in many songs just to go with the melody.
Because Japanese tones vary depending what region you're in (different regions carry different dialects, with varying pitches for the same words), it's not practical to put it in a song that would want to be played in all of Japan.
Taken from wiki (https://en.wikipedia.org/wiki/Japanese_pitch_accent): "For instance, the word for "now" is [iꜜma] in the Tokyo dialect, with the accent on the first mora (or equivalently, with a downstep in pitch between the first and second morae), but in the Kansai dialect it is [i.maꜜ]. A final [i] or [ɯ] is often devoiced to [i̥] or [ɯ̥] after a downstep and an unvoiced consonant."
As far as I know, pitch accent is less relevant in Japanese pop music than it is in English pop music partially because Japanese syllables are more percussive than English syllables are (think: one consonant-sound and one vowel for (most) Japanese syllables), and so sound less unnatural when pitch accent changes for the sake of a melody.
However, sometimes, guttural stops such as words with (small) っ are sometimes pronounced as enunciated extra vowels with stops in between: A word such as きっと "kitto" might be easily heard as "ki-i-to" instead.