There's no problem with double-clicking while playing a game or something, but there is while trying to run an application using a double-click. It works sometimes but mostly not. I have to use right-click and then open or start clicking LMB very fast. I've got double-click speed set properly, so it's not the problem. My mouse is brand new. Got drivers updated. The problem occures without drivers too. Any ideas what's wrong?

Try double clicking very hard. Press LMB very hard each time you click.

You've said clicking LMB very fast works sometime. That means double-click speed is not set properly but you've also said double-click speed is set properly which is confusing. Double check the, double clicking speed! In the mouse properties where you set the double click speed, adjust the slider and test double click speed in the test icon at the right. If it works with the test icon it should work anywhere.

Most games over-rides operating system's mouse settings and use their own so there is a possibility that it works in games and not in windows.

Pressing LMB very hard works, that's what I am doing when I can't double click (clicking very fast works too but not that good as hard, single clicks).
Double click is set properly because only most of the time the double-click doesn't work - sometimes I can double-click as usual, in pursuance of double-click speed. The double-click speed is set properly not only in Razer drivers but in control panel too.

I guess its time for a new mouse. Buy a robust one. Yeah, I know the one that you are using now is a new mouse, but some are very fragile. And when you play a lot of games, they don't last much

Well, umm... if it was some ****ty noname mouse I'd listen to you but it's Razer Deathadder, there is no way that it could break just like that. I've still got a warranty though.

Yeah, it can...that's what the warranty is for.

Whether its a Razer DeathAdder or a Cheap LifeSaver, its subject to failure. If its under warranty RMA it, thats what you paid those extra bucks for. If you are an enthusiast and willing to take logical risk, study this guide and fix it yourself,
http://forums.hardwarezone.com.sg/showthread.php?t=1793115

CuratoR,