Firefox has a built-in setting (about:config item browser.tabs.insertRelatedAfterCurrent) to automatically open tabs next to the current tab, but it only works when clicking a link inside the current tab, and will then open that tab next to the current one. But if you click the New button or press Ctrl+T, it will always add the new tab at the end of all the tabs.

To change the the way that New-Tab works, install the add-on Open Tabs Next to Current.

A non-add-on way is to use the bookmarklet described here:

javascript:window.open('about:blank');void(0);

Click the bookmarklet in the Bookmarks Toolbar or assign it a keyword to open a new tab after the current one.

Here's how to create your own keyboard shortcut to Detach a Firefox Tab to a New Window:

1. Install the Keyconfig addon

2. Open the Keyconfig Preferences by pressing Ctrl+Shift+F12 (On Mac it may be ⌘ Cmd+Shift+F12)

3. Click Add a new key at the bottom left of the keyconfig window.

4. Replace the contents of the Name field with the name of this command:

   Detach Tab to New Window
   

5. Paste the following code into the /* CODE */ field:

   window.openDialog("chrome://browser/content/browser.xul","_blank","chrome,dialog=no,all",gBrowser.mCurrentTab);
   

6. Click OK.

7. Your new command will be marked as <Disabled> This means you need to set a keyboard shortcut for this command. Make sure your command is selected from the list of commands. Click inside the field at the bottom left of the window. It will highlight the contents and whatever keys you hit next will be entered inside that box. For example, you might use Ctrl+Alt+N. Or if you're a fan of Vim you might use Shift+Y.

8. Click Apply and then click Close to close the keyconfig window. Test your keyboard shortcut by loading a window with at least 2 tabs and then use your new keyboard shortcut. The current tab should detach itself and form a new window.

Actually i would like to comment @avirk's answer, but, i don't know why, i can't comment any answer to this question.

@avirk's answer helped me to completely solve the issue, so I give him the bounty, but i have to change some details, and so, I want to summarize all the actions that does the trick for me, for the future readers.

• Оpen page about:config, and change the value of variable browser.tabs.loadInBackground to false;

• Install Tab Mix Plus addon;

• Open Tab Mix Plus settings;

• Go to Links tab and set the option Enable Single Window Mode

• Go to Events tab, Tab opening sub-tab, and in the group Focus/Select tabs that open from set the option Links.

• At the same tab and sub-tab, in the group Inverse Middle-click or Control-click focus of: set the option Links.

Now, the problem is solved: Shift+click opens link in the new tab and moves focus to new tab, and Ctrl+click opens link in the background tab. Exactly like Opera does, which is needed.