I think the poll question is too broad and nebulous to offer a meaningful answer. However, the specific questions are worth addressing (though some of them are pretty broad and subject to interpretation, too).
Is high randomness...more fun?
Depends on what you consider high randomness. I like some degree of randomness, but I also like problem solving and meaningful choices in the RPG. Also depends on where and how the randomness is being applied.
Is...convoluted randomness...more fun?
Convoluted has a somewhat negative connotation, implying needless complexity. Thus, I'd say no, needless complexity is not more fun.
Is...random randomness good design for an RPG system?
When considering random randomness, I suppose it contingently depends.
For instance, instead of a set 10% chance for something, is it better to roll 1d20 to see what the percentage chance is (and then roll the d%)?
If I think the chance should be about 10%, then I'd just set it to 10%. If I don't know/care or want a more random spread, I might roll a die, but I think that would be rare. If I'm writing a game supplement and want to suggest an appropriate range, rather than a discrete percentage chance, I might list it as 1-20%.
Is it good to have a series of charts where rolling on one determines the next chart to roll on?
Sure. I find that kind of thing useful. (Although it depends on the place the series of charts is used. If it's in the middle of combat, maybe not...)
Is it fun to have a system where a result can be from fantastic to terrible (like "character gains a level" to "character dies") -- like drawing from a deck of many things?
Not always, but it certainly can be. I like to present this kind of thing and leave it up to the players whether they want to risk it or not. (Usually somebody does.)
If rolling randomly (straight, no drops or rerolls) for ability scores and hit points is good, is rolling for starting level even better?
Not necessarily. But it might be appropriate in some circumstances.
Is a randomly generated dungeon -- random rooms, corridors, monsters, treasure -- more fun than one put together with a plan?
No, I don't think so. However, I do use random tables and such to help me flesh things out, and also as a springboard for my own creativity. I think random tables (e.g.
the Monster & Treasure Assortments) and lists (e.g. the cool stuff in the 1e DMG) are useful tools.