Tested more "Microsoft oriented" applications beside Skype - OneNote, Skydrive, and they all have same error "Cookies must be enabled in your browser" when I try to login to MS servers with any of them (and cookies ARE enabled in browser).
When I revert back to stock ROM, all apps work without problem.
Same problem is described in threads for codefireX ROM for Desire HD, but I can't find a solution on xda, or on the web.
I've even tried to change permissions to 777 for /data/data/com.andorid.browser/databases folder and databases in it (webview.db, webviewCookiesChromium.db, webviewCookiesChromiumPrivate.db, where cookes are supposed to be stored), but it went without a result
Does anyone else know a solution for this issue ?