How shoud I properly initialize DApp and get account address?

Question

I am writing DApp in react, and basically I want to initialize web3, unlock account, and display it. Having web3 in my package.json In the onload I've tried to go following:

let Web3 = require('web3');
let instance = new Web3(window.web3.currentProvider);

First of all, I am completely missing the point of such a step. What do we do here? What is the difference between window.web3 and instance now?

Second, what line should I use and why?

console.log(window.web3.eth.accounts[0]); // this?
console.log(instance.eth.accounts[0]); // or that?

I want to figure out browser-agnostic scheme, that would work on Metamask, Opera, and on mobile browsers like Cipher. I've tried a bunch of schemes, on some platforms I'm missing web3.eth.accounts[0], but accounts can be availble via web3.eth.getAccounts()

I've looked at following, but still can't generalize:

• Better pattern to detect web3 default account when using metamask
• New web3 method to request account access from dapp browser, no request

I run into a problem that it's neither straightforward, nor documented process. To face such a problem, lets create a place with answer, detailing basics of init and unlock procedure, that should work in any browser, for any web3.