I have some scripts (AutoIt) browsing YouTube for list of trending videos etc. It involves no mouse clicks (just keystrokes). The script takes a long time to finish and I can't use my PC during this time (it needs the window activated to work on it).
Is there anything I can do about this? Can these scripts run from a server or some stuff like that?
As Johannes said, AutoIt probably wouldn't be suitable (and likewise AutoHotkey), but you could check out the many GUI and web testing frameworks that exist for other languages. With some of those, you can run a "headless" browser (a program that navigates the web just like a browser, but has no visible window); or you can run a standard browser on a virtual display like a Xvfb X11 server. This would be easiest if the server (or wherever it's going to run) is running a Unix-like OS, but it may be possible with Windows too.
Selenium Webdriver seems to be a very popular choice for scripting and testing actual browsers. It's natively Java but has bindings for languages like Ruby. It can also hook into something called HtmlUnit, which is also Java; that's one of the more popular headless browsers. Another (a relative newcomer) is phantom.js, which is in Javascript but (again) has bindings for other languages.