Your question is very broad. I voted "no," but I'm sure that what I see in your question is influenced by my own opinions and tastes.
I am an unabashed problem solver in games. I like to come up with ideas and to make strategic decisions in order to accomplish goals. Randomness, particularly in combat, can make this more interesting for me as long as the randomness isn't too extreme. By making combat involve many small decisions, and then randomizing the success of most of them, it gives me the opportunity to make tactical decisions and adapt to the success or failure of my previous efforts.
Meanwhile, I loathe high degrees of randomness. I am an elitist on this issue, and I won't apologize for it. Not only does it frequently stop me from effectively planning or making strategic decisions, but its... mindless. I get no particular pleasure from rolling a 20, and no feeling of failure from rolling a 1. The odds of those things occuring were the same both before and after I actually rolled the dice.
Some people like to talk about the drama of sudden success or tragic failure that is generated by "swingy" combat. I don't experience that drama. To me, its like talking about the drama of a lottery ticket or the game Deal or No Deal. There is no drama. Its purely mechanical. No one has any right to be proud of their success or to rue their failure.
This isn't to say that no RPG should have high degrees of randomness. Some RPGs aren't about problem solving in the first place, or, they put the problem solving somewhere other than where they put the randomness. For example, Paranoia involves plenty of problem solving, the problems you solve are more about how to fast talk Friend Computer while screwing over your allies, and that's not randomized. Og: Unearthed involves problem solving, but its all about language and communication, so the highly random combat doesn't affect it. And Feng Shui really isn't about problem solving, its about badass combat description. So randomness is fine by me in those contexts.
But in D&D where a big part of the game, at least for me, is "how can I and my team defeat the orcs that just attacked us?" I want random factors to spice things up, not negate the "how" in that sentence and turn everything into a mechanistic dice fest.
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%)?
That sounds mathematically pointless. Maybe I'm misunderstanding, but if you have a randomized list of random chances, you can just multiply through and come up with the REAL random distribution, and then assign dice as necessary to accomplish it in one roll.
Is it good to have a series of charts where rolling on one determines the next chart to roll on?
Gods no. I've played enough Battletech to know that this just makes things take longer without adding much. Again, any time you have one chart seeding you to another chart, you COULD just make one chart with all of the information. Or you could come up with a less chart based game, reducing the amount of time spent looking up data... that works too.
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?
Its not fun for me. I concede that many others seem to enjoy this. I don't understand why.
If rolling randomly (straight, no drops or rerolls) for ability scores and hit points is good, is rolling for starting level even better?
Rolling randomly ISN'T good. Its that sheer gambler's thrill that I look down upon as beneath me.
Is a randomly generated dungeon -- random rooms, corridors, monsters, treasure -- more fun than one put together with a plan?
Well, that's a little different question. Its probably less fun, particularly in an RPG context. But it has its place when you just want to take a character and trash a dungeon, and you don't have or want to spend prep time.
