My ultimate favorite IDE = Netbeans
It's very plain, the more you use it, the more you learn it. Just hit open and open you html files, or drag&drop them from your desktop and you ready to code. Or, start coding right and put your files in a project and let Netbeans handle it for you. Moreover, the editor is customizable. You can change font, background colors, size etc. Got good community support and is Open Source.
checks my HTML code for errors: - Got it and very well done + it formats your HTML codes and show tags in colors.
just by clicking some "run " button I am able to see the output: it got it's own integrated browser in a separate panel to display the result + can be linked with your favorite browser e.g. Firefox, Chrome. If you hit run Firefox will fire-up and run your html page.
runs on Windows and Linux: Checked. Win + Mac + Linux.
it should be free and not complicated to use for beginners:
Yup, Free. For beginnings? hmmm, means no complicated stuff, so don't use the advance features available. Netbeans can take you from newbie to pro, the more you use it the more you will learn about it which can save you lots and lots of time in the future. (Let's say you want to change all
<div> similar to notepad++, you can find&replace using text, regex, wild card etc, you can change in files/directory/project)
IMO The best way to learn is to code it yourself. IDEs that allow you to use visual means to build your HTML pages are not very good way to master HTML. Netbeans can work as an easier notpad++, I appreciate even small stuff, like if you CTRL+LEFT CLICK on a url in one html file and it will take you to that source code of that url html file.
You can use it as a one time quick edit IDE for a single page like notepad++ or in a project based where you compose all of your files under one roof/directory/project/namespace etc. as you call it. It allows opening multiple tabs, same or different panel/windows, different colors for tabs based on to which project they belong to. Plus, open a page in a project by just typing it's name, search for file/keyword in project, navigate from project to file or from the opened file to it's base project.
It has a smart html tags checking, highlight errors, show hints, auto complete, HTML drag&drop panel. I prefer it for learning since it shows your mistakes and give you tooltip to know what do add. Type
<html> and hit CTRL+SPACE and it will auto close it for you by adding