Starting a Program with Powershell Remoting

Question

Using Windows Powershell, this opens notepad.

notepad.exe

But this, which is meant to remote into a virtual machine and open notepad there, does not. Instead, the process starts but the GUI is hidden.

$s = New-PSSession -ComputerName MyVirtualMachine
Invoke-Command -Session $s -ScriptBlock {notepad.exe}

I need to know a few things.

1. How do I start a process with Powershell remoting such that its GUI is visible?
2. Why is the GUI invisible?
3. If I follow your advice regarding item 1., what are the implications regarding how the process starts? I want the process to start as the user I am logged in as (same as credentials I am connecting Powershell with).
4. I want to choose whether the process has administrator privileges (the credentials will) as I need to run tests on processes both with and without admin rights.
5. Other factors you can think of which may affect how the process runs. I am testing software, so the behavior ought to be identical to a user double clicking on an icon.