The dream not having a plot seems to be common in the people that I meet. Dreams are supposed to be random and not make any sense. That's fairly constant. Not sure if the shifting plot lines is common, but the random dreams are.
Being other people in my dreams... Not that's not that common. I get odd looks when I tell people that one. I feel like I have been all genders, ages, races in my dreams and all with different situations. One particularly vivid dream involved me being a nameless Boy Scout who had to break away from his troop to pee on a tree (and yes, I was a boy while I was doing this so... that was interesting). The only other people who have experience this that I have met have been writers. This has led me to believe that the two are interconnected, that the reason I experience this because all day, I create characters and/or further develop them. I have to get them in my head and often have intimate knowledge of them. So, of course, naturally, this extends to my dreams. A lot of the time, this means doing my best to make it perfect, discarding or changing all of the details that aren't quite right. I have wondered if my dream characters are my throwaways.
Of course, it is rare for me to remember a dream as it is for most. Most will forget their dream within a day, maybe a few minutes. Some people make it a point to write down their dreams in a dream journal and, while this is admirable, I have two reasons I can't do this: 1) my brain can't think when I first wake up and 2) I barely have enough time in the mornings as is. How could I fit in doing a dream journal, which would at least take twenty minutes?
Some people dismiss sleep and dreams as an REM process, our brain just making sense of the day. Most don't think that it means anything. Yet others really make it a point to analyze and remember their dreams. There is even a website called dreammoods.com devoted to dream interpretations that I frequent from time to time and some seem quite random (dreaming of canned foods means bottled emotions?). I try to analyze my dreams and some I can, but most I can't make sense of.
A percentage of the population can even experience lucid dreams. Essentially, lucid dreaming is active dreaming, that is being able to know that you're dreaming and manipulate the events in the dreams. This can be a time to act on fantasies you wouldn't be able to act on in real life or flexing one's creative muscles. Apparently, this can be accomplished through hypnosis and the keeping of aforementioned dream journals. I have tried hypnosis, but it hasn't worked for me. This is a shame, because lucid dreams sound like a lot of fun. But oh well.
Dreams are what makes sleeping fun, though. Nightmares are never fun, naturally, but other dreams are. Often, my dreams can be very pleasant and are puzzles for me to figure out. In my most tired moments, I don't dream, but that's no fun. It's always cool to get so stuck in my dream that it puts me in a deep, deep sleep, because that usually makes it more enthralling. Sleep would be so different without them.