Can apps purchased for iPad be downloaded on MacBook I purchased the iA Writer app for my iPad. Can I download it free on my MacBook? 

Thanks.
You can download it, for the purposes of backup/syncing etc with iOS devices via iTunes, but it is not an OS X application so it will not run.  Some companies release software for both platforms, often with the same name/version number/price point, but they are separate applications, and when you buy you are purchasing on a platform specific basis, so even if you own iA Writer (or Byword, or any other cross platform app) you only own it on the platform for which it was purchased.