I don't understand what you mean here. It sounds to me like you're trying to use a smartphone like a desktop PC. There is absolutely zero need for html5 based web apps on an Android smartphone. There's a native app for every thing you could possibly want or need (more or less).
Installing a web app won't install Flash. Is that why you've been after flash? For web apps? Then you definitely don't need flash as you don't need to run web apps on a smartphone. Want gmail? Use the already installed gmail app. Want some other webmail service? Use the Email app included with the phone. Wanna access YouTube? Again, use the app.
Personally, I feel most of these browsers are overhyped and overrated. The default browser is all you need for your basic browsing and unlike on a desktop computer your not likely to live in your web browser as everything is app based. I'm not talking web apps either.
I think what you're talking about is "Firefox HTML5 Supported" which is from some random dev and seems to be a niche product that likely wouldn't be of any real use to you. Android apps come with everything they need. Yes, Firefox for Android has minor extension support but as I said, you shouldn't be living in your browser so extension support isn't really needed. If you REALLY want an alternative browser to the default though I've heard good things about Dolphin Browser.
Again, remember, this is a smartphone and not a PC so things (the ecosystem in particular) work rather differently.
