How can I make Windows Server 2012 R2 like Windows 8.1?

Open an elevated PowerShell and run:

Add-WindowsFeature Desktop-Experience

among other stuff, this will install the Windows store which lets you download other modern apps.

I'm not sure whether this solves all your problems, but it is a start.

You will find some of your questions answered in the article :
Convert your Windows Server 2012 to a Workstation!
which contains many more adaptations than you have asked for in your post.

The section Windows 8 UI Internet Explorer especially shows how to make Internet Explorer run as Windows 8 UI application when started from the Start screen. As this involves registry changes I have not replicated it here.

The problem of using an Online Microsoft Account (Windows Live authentication) is still unsolved in that forum.

That said, I would second the comments that advise against using Windows Server 2012 R2 on a laptop. It would be more preferable to run Windows 8 with Server 2012 as a virtual machine, which would give you a better user experience.