Normally we will facing Sql Server problem if we change computer name after installing Sql Server.
How to solve this problem? Just run script at below

sp_dropserver <'old_name\instancename'>
GO
sp_addserver <'new_name\instancename'>, local
GO

Further more, please refer to How to: Rename a Computer that Hosts a Stand-Alone Instance of SQL Server.

How to install SQL Express using Visual Studio Installer Setup Project?

Here some hint:

1. Use Commit node at Custom Action Editor
2. Add a process to call batch file that will call Setup.exe for SQL Server Express.
3. Or if you don’t want using batch file, you can create another .net application that will call setup.exe for SQL Express.

Note: You can’t directly call SQLExpress Setup.exe from Commit node at Custom Action since Setup Project not allow recursively call msi installation.

Reference: SQL Express Silent Installation with .Net Setup Custom Action

Beside that, still have other way to install SQL Express too. Examples below will install SQL Server during first time you application start.

• Install SQL Server (Express) with your Application
• .NET Installer that Automatically Installs SQL 2005 Express
• Embedding SQL Server Express into Custom Applications