Q:

Is browser programming standardized and do all major browsers conform?

This is more of a curiosity question, to see how things have evolved, as I do not currently do any kind of web programming, but I remember when writing JavaScript was something like this (pseudocode):
if (foo exists)
  call foo
else if (fooAsFirefoxNamedIt exists)
  call fooAsFirefoxNamedIt
else if (fooAsOperaNamedIt exists)
  call fooAsOperaNamedIt
else
  message "We are sorry, you have Internet Explorer"

Has the dust settled? looking around it looks that there is a standard ECMA-262. Is that right? If so, do all major modern browsers implement a standardized JavaScript?

A:

There are two major aspects to this:

Has JavaScript the language, stabilized, and
Have the browser features provided to JavaScript (such as addEventListener vs. attachEvent) stabilized.

Has JavaScript Stabilized?
JavaScript was far too stable for years. :-) ECMAScript3 was released in 1999; we didn't see the next update (ECMAScript5, there was no released ECMAScript4) until a decade later.
Other than antiquated browsers (IE6, IE7), JavaScript features were stable (other than Array#indexOf, I think) for a long time until ECMAScript5 arrived. Then, while most browser vendors brought their browsers into full compliance fairly quickly, Microsoft dragged for years, and so you saw and still see a lot of shims/polyfills for Array#forEach, Object.create, and such because people had/have to support IE8, which came out just before ECMAScript5, and it was years until IE9 came out.
Sadly, though, IE8 remains in significant use thanks to being the latest version available for Windows XP, which itself despite being end-of-life'd remains in significant use.
The next version (ECMAScript6 aka ES6) is due "soon" (some are optimistically calling it "ECMAScript 2015"), and naturally because ES6 has some very major features in it (it's easily the biggest update JavaScript has had), those will come in in dribs and drabs across vendors' browser releases. In the meantime, many though not all of the features can be used today via transpilation, where you write code in ES6 and then transpile it to ES5. If you search for "ES6 transpiler" you'll find several.
Have Browser Features Provided to JavaScript Stabilized?

looking around it looks that there is a standard ECMA-262. Is that right?

That's correct for JavaScript (above), but for other things, it's much more the HTML5 specification, which goes well beyond just HTML. It's also the various DOM specifications.
Things are a lot better than they were five years ago, for two major reasons:

Browser vendors are working together more than ever before in terms of establishing standards for what browsers should do, from the File API to web workers to web storage and a dozen other things.
Microsoft has made a serious effort to bring IE in line with standards. IE9 is a dramatically better browser than IE8 in terms of standards compliance, IE10 is better still, and IE11 is better still.

Some useful references:

caniuse
kangax's ES5 compatibility tables
kangax's ES6 compatibility tables
kangax's ES7 (yes, really!) compatibility tables

