Nope, the desktop application doesn't show a due date for the actions in the way that the iPhone app does, you just get the due soon or overdue styling. It does use the implied due date from the project when sorting, although there's a bit of a gotcha -- it doesn't do the same for subsequent actions that might have a due date if different than the project. If I have 5 actions in a sequence, no due date on the first two, and a due date of Wednesday on the third, one might expect that it would treat the first two as having a due date of Wednesday for the purposes of sorting (as it would do if the project had a due date of Wednesday) but that's not what happens.
However, it's pretty easy to set all the due dates for those actions to be the same as the project's due date (unlike on the iPhone). Click on the project name in the sidebar, switch to the outline pane via cmd-4, click on the first action, shift-click on the last action, bring up the inspector, and put the due date in.
If you want the desktop application to behave like the iPhone app does, send in your vote with Help->Send Feedback or email
omnifocus@omnigroup.com.