i dont exactly know how it works in2010 with the wsp but i would of thought that the initial wsp can be converted which is fine as it uses .net 3.5
the issue is your getting the error:
Could not load type 'assemblyname'.
this tells me that there is a reference that is not referenced! so in your project within vs you need to upgrade custom dlls that are referenced for 2010 to 2013!
as noted:
Visual Studio 2012 -> Create new solution -> Under SharePoint 2013 select "Import SharePoint 2010 Solution
that should solve the issue.. but as you are looking for it to work off the bat i would presume that a dll within the project under reference is not set properly or doesnt exist in 2013 to work but it does in 2010!
have you tried a simple webpart like hello world? click a button and it shows hello world on a lable? see if it works which it should. Im still going with the fact that there is a dll in your project that doesnt exist in 2013 (assemble for 2010 3.5) and not assembly 4.0!
EDIT
ok this is worth a try:
try putting the dll that is causing the issue within the web application dll folder!
C:\inetpub\wwwroot\wss\VirtualDirectories\youwebappFolder\bin
that should do the trick ;)
just to make it clear that the error your getting is Could not load type 'assemblyname'. ? so there is somthing called assemblyname in your project? or have you just renamed it?
if the issue still persists than you need to force compile on the .cs class that you have named 'assemblyname'. within vs2012 goto the .cs class in question -> properties -> change build action to compile! build and deploy, the 'assemblyname' should now work as its compiled properly within the dll and packaged properly within the wsp!