Can I create a block that shows custom node types but in a specific order?
You can use Node Queue module, it may solve your problem.


  The Nodequeue module allows users to collect nodes in an arbitrarily ordered list. The order in the list can be used for a any purpose, such as:
  
  
  A block listing teasers for the five top news stories on a site
  A user’s favorite music albums
  A group of favorite from which one is randomly displayed
  
  
  Nodequeue provides a simple drag-and-drop interface to manually order any queue. Additionally, it allows nodes to be added and removed from queues without needing edit permissions to the node. Nodes can be added to queues either from a queue management tab or by links on the node teaser.