My view:

Short answer: it's suffering that's bad, intrinsically (though suffering can be instrumentally good)

Long answer: There are several different reasons suffering may be voluntary. To list a few:

1) suffering for some greater good (eg delayed pleasure, suffering for something that will make more people happy, etc)

2) false belief that your suffering is for a greater good (eg you think suffering will give you karma points that will make you happier in next life)

3) suffering that is "meaningful" (such as mourning)

4) an experience that includes some suffering and some pleasure that is one the whole-enhanced by the suffering

For 1, the good that the suffering leads to is intrinsically good, the suffering is instrumentally good but intrinsically bad. If you could get the greater good without the suffering, that would be better.

2, 3, and 4 are really just special cases of 1. For all, the suffering component of the experience is intrinsically bad. For 2, you falsely believe the suffering is still instrumentally good. For 3, the "meaningfulness" of the experience is the greater good, and the suffering is instrumental in that. It would be better if you could get the same amount of meaningfulness without suffering. Similarly for 4 - the pleasurable part of the experience is the greater good.

I'm a hedonistic utilitarian, and I think that even voluntary suffering is be intrinsically bad, as long as it's still suffering at that point. Here the reasons I explain the phenomena that you note in your question. My answers are partially overlapping but some of the solutions you suggest.

- I personally mostly listen/watch/read media that deals with negative emotions. When I do this I sometimes have a twinge of the negative emotion, but I don't think I would really describe it is negatively valenced. Sometimes it may involve a bit of negative valence, which may be outweighed from the aesthetic appreciation that I get from it.

It seems like 'negative emotions' can sometimes not have negative valence in this way even though they retain their other features. I think this is similar to how 'pain' from exercise can sometimes have a neutral or even positive valence (at least that's how I characterize it). It seems like the secondary resistance to emotions can be generating some or all of the negative valence associated with them.

- Similarly, for an emotion like grief, I think it's either the case that I don't experience it as really negatively valenced or I'm getting immediate counterbalancing positive emotions from it (like a sense of meaning and connection).

- Sometimes negative events can be cathartic, meaning that they provide relief from the negative emotion. I often find crying to do this and crying sort of feels good for this reason (or at least it feels much less bad than the alternative in that situation).

- I think sometimes I also irrationally pursue negatively balanced emotions. For example, by ruminating. Not sure that I have anything insightful to say about why this happens.

It seems to be hard to figure out exactly which of these is happening in a given situation.

Buddhism would say that if you experience sadness without craving that the sadness go away, you continue to feel sadness but you don't suffer from it. This corresponds to my personal experience. There can actually be richness in the sadness that I enjoy. I know that many other people enjoy it too because there are so many sad songs and movies. When something sad happens to me, I try to prolong it as it is a pleasant and positive experience for me. I think that a Buddhist would say that this is bad as well because I feed a craving and the goal is to get rid of all cravings. But I think it's no worse from the Buddhist perspective than trying to prolong a happy experience. However, I noticed that in the past I (not fully consciously) subtly caused some bad things to happen out of my desire to feel sad. I guess you should look out for that if you start enjoying sadness too much. The things I was doing were bad for me from the long-term perspective.

In contrast, I haven't yet conquered guilt, remorse, and jealousy. When I feel these emotions, I suffer and want them to go away. When a relatable character in a TV show does something predictably bad or cringe-worthy or embarrassing, I hate it and turn off the TV because it causes me suffering. Most people feel more comfortable with these emotions but less comfortable with sadness.

I'm a bit confused about depression though. When you are depressed, maybe you don't want to be happy because you don't remember what it's like to be happy anymore? Or maybe you want to experience calm positive emotions, you just don't want to be artificially cheerful?

My intuition is that suffering is bad, but sometimes (all things considered) I prefer to suffer in a particular instance (e.g. in service of some other value). In such cases it would be better for my welfare if I did not suffer, but I still prefer to.

I also think that in cases where one voluntarily suffers, then this can reduce the suffering involved. Relatedly, I also imagine that voluntarily experienced pain may lead to less suffering than coerced pain.

It also seems to me that there are cases where we directly want to experience a suffering-involving experience (e.g. watching tragedies and wanting to experience the feeling of tragedy). I think in many of these cases the experience is sad, but also involves (subtle) pleasures and what we want to experience is this combined set of emotions. In some such cases I'm sure people would prefer to experience the distinctive melancholy-pleasure emotion without the suffering valence if they could (but cannot imagine, let alone actually achieve this), and in other cases people would not with to detach the suffering from the emotion set (because they have preferences to have fitting responses to tragedy and so forth). I am sure there are a whole bunch of other factors which explain people propensity to voluntarily watch tragedies though e.g. affective forecasting misfires, instrumental goals like signalling, and feelings of compulsion (tragedies tend to be very salient and so adaptive to pay attention to, even if they entail suffering).

I think we don't quite have the words to distinguish between all these things in English, but in my mind there's something like

• pain - the experience of negative valence
• suffering - the experience of pain (i.e. the experience of the experience of negative valence)
• expected suffering - the experience of pain that was expected, so you only suffer for the pain itself
• unexpected suffering - the experience of pain that was not expected, so you suffer both the pain itself and the pain of suffering itself from it not being expected and thus having negative valence

Of them all, unexpected suffering is the worst because it involves both pain and meta-pain.

Since many other answers treat the more general ideas, I want to focus on the "volontary" sadness of reading/watching/listening sad stories. I was curious about this myself, because I noticed that reading only "positive" and "joyous" stories eventually feel empty.

The answer seem that sad elements in a story bring more depth than the fun/joyous ones. In that sense, sadness in stories act as a signal of deepness, but also a way to access some deeper part of our emotions and internal life.

I'm reminded of Mark Manson's quote from this article:

If I ask you, “What do you want out of life?” and you say something like, “I want to be happy and have a great family and a job I like,” it’s so ubiquitous that it doesn’t even mean anything.

A more interesting question, a question that perhaps you’ve never considered before, is what pain do you want in your life? What are you willing to struggle for? Because that seems to be a greater determinant of how our lives turn out.

Maybe sadness and pain just tell us more about other and ourselves, and that's what we find so enthralling.