How can I access files on a Seagate GoFlex Satellite wifi or Kingston Wi-Drive? My final goal is to programmaticaly access files on a Wifi/USB storage device, like a Seagate GoFlex Satellite wifi or Kingston Wi-Drive (preferred because less expensive).

The storage devices are made to be connected to a PC by USB and also can be connected to Android (and iPhone) by wifi.

Once connected, it is possible to put files on the hard drive over wifi using a custom Android application provided by the manufacturer (like Kingston Wi-drive or Seagate GoFlex Access™ app)? How do they do this? Can I access on my filesystem with program like Astro ?
According to this review Pocket-Lint: Kingston Wi-Drive review :


  You are also able to access the files on the Wi-Drive on a PC or Mac
  by visiting it's IP address in your web browser. This, again, allows
  you to play any files that are supported in your web browser. MP3s,
  for example, play fine, but you might struggle more with video files.
  
  You can also connect to the Kingston as a regular network drive. We
  were able to view it as if it was any network attached storage device.
  That means we could also copy files over wirelessly too.


This suggests that, the Kingston at least, supports something like SMB (the Windows file-sharing protocol). Many Android apps let you connect to an SMB shared drive to copy files on or off, I use File Manager to copy files from my Windows machines to my phone/tablet this way, and I'm pretty sure Astro supports it too.

To do this from your app, you just need to include SMB file access support in your Andoird app. How to do that is off-topic for this site, as that is a developer question.