Can I install SQL Server Express 2008 on a Windows Server 2008 or should I stick with the SQL Server Express 2005?
On the download page for SQL Server 2008 (with tools), it lists these supported operating systems: Windows Server 2003 Service Pack 2; Windows Server 2008; Windows Vista; Windows Vista Service Pack 1; Windows XP Service Pack 2; Windows XP Service Pack 3
Yes, it runs without difficulty and is painless to install, especialy if you use the Web Platform Installer, which can also install Management Studio Express to help administer the database.
Yes, you can. The Hardware and Software Requirements for Installing SQL Server 2008 lists all of the different versions of Windows Server 2008 supported.
Why would you assume this to be impossible?