Are there any copyrights on the Font?
Just like any professionally produced font, you can't just use it without getting or buying a license. Ecofont is a bit different:
There is Ecofont software, a whole printing software. You won't have to typeset your documents in Ecofont, you just have to print them using the software.
Actually, in most jurisdictions, fonts are treated like software, so you buy licenses to use them. Depending on the size of the company, the number of workstations where it will be installed, the organization type, you will receive a quote by Ecofont.
You can also use a demo version. It includes an option to buy the license.
You can use the normal "Ecofont Vera Sans" typeface which doesn't require any software and should come for free.
Are there any better, more economic Fonts like Ecofont?
See the linked question.
Is it easy to install it? How can I do it on WinXP/Win7/Ubuntu/Debian/Fedora/etc.?
The fonts themselves, just like you would install any other font that comes as OpenType or TrueType. In Windows, just right-click and install. In Ubuntu and similar, there's a Wiki. There are of course resources for Fedora too.
Since Ecofont actually is a printing software package that offers more than just the font, they have a FAQ if you encounter any issues. This software runs on Windows, OS X and Linux.
Does it really save 10-20% ink?
If you trust their information, yes. But you'll have to read reviews.
Does the printed text has the exact same lifetime? (vs. a normal font) doesn't Ecofont wear out faster compared to a normal font?
Normally yes, but it rather depends on how good your printers are (inkjet vs. laser), what kind of paper you use (heavy vs. normal), but not on the font. Given that the font uses less ink, it might expire faster if your printer produces bad quality prints.
Are there any websites that has a lists of companies that changed the "default font" to something like Ecofont?
On Ecofont's website itself: