The developer responded to my emailed feature request (specify a default due date) within one day to say it's on the list for a future release. Only glitch is that synced tasks created in TTDLP show up on Toodledo.com with an "=" sign in the due date box. It doesn't affect sorting, just looks cluttered in the web interface. I reported it and hope it gets fixed in a future release.
I discovered that when adding a task, you can type in "today" in the date box and have it fill in "Today" rather than having to click on the calendar or the "Today" link. It's not as quick as being able to type in "t" and have it autofill, but I've been looking for a way to enter new tasks without touching the mouse and this allows me to do it for tasks that are due today.