I'm very curious if most instruments play a melody at the same time, and work in counterpoint, or it's just a singer playing a melody while some instruments play chords.
It's often hard to draw a line between melody and chords.
I know, that sounds ridiculous. But sit back and think about the role a keyboard plays. Sure, they can just play a single note melody, or they can just hold some chord tones. But often they do both, at the same time. They voice the chords they play, so that a melody is formed. Sure, it's not the main vocal melody, but it's a melody nonetheless.
This happens with other instruments too, but I'm a keyboard player, so that's what I'm most familiar with.
Even a bass line can be a melody. In fact, thinking melodically is often a mark of a good player. Sure, taste is important here; it's easy to be distracting, but bass should be more than playing quavers on the root of the chord.
So, to answer your question, the instruments of the play both chordally and melodically, and will often change role during the song. Even the vocals might be chordal (if there's more than one singer). You can deconstruct specific examples, but it's difficult to generalise across a genre, especially one as broad as pop.
Yes, a lot of pop music is guitar-based. You feel it could have been written by a singer simultaneously strumming a guitar. A bass guitar might add an element of counterpoint, as might a lead guitar. But basically the song could be performed by just a singer plus guitar playing chords.
But a lot of pop songs do other things. The musical background may be a repeated 'beat' with no particular chord sequence. Or it may be composed and scored in a 'classical' style. 'Pop' covers a huge range of styles!
But what a strange question to ask. LISTEN to some pop songs. What musical methods DO they use? Don't ask us, use your ears!
There is not a 100% answer to this one. I have seen a lot of pop music with examples of both. I think that pop music is heavily defined by vocal melody and rhythm, though. It's less about the pitched instrumentation these days. Seems like the formula people are gravitating too depends on if the girl can sing, and the song has a hip beat.
It's difficult to answer whether "most" pop songs do anything. I can say this though, one interesting characteristic of pop music designed for the radio is that the beat of the song is recognizable almost instantly. Meaning that for most pop songs you can tell what song is playing simply by listening to the instrumental without the vocals (there are many reasons why producers put so much emphasis on this).
Now there are many ways how producers make their instrumental have a unique character that'll allow for instant recognition: one way is by containing interesting or unique sounding instruments. Another way is by having the instruments play a certain melody or motif that revolves either based around the chords creating a "half chord progression, half melody" or based around the vocal melody usually repeating the hook of the chorus (e.g in the songs Symphonies by Zara Larsson or One Last Time by Ariana Grande) this usually played by a guitar or piano line but nowadays it can be almost anything that plays the "motif").
To conclude: there isn't one rule that all producers follow, but it is common for producers to include a melodic phrase/motif/loop in their instrumentation as a way to make the beat stand out. (there are other reasons as well but I think that would be a bit too deep... Let's keep it light shall we)