Tupelo,
I rode it in May again with my husband, so not solo, but I don't think you would have any problems safety wise riding alone. Doing it in 3 days imo would be hard and require you to ride it at such a pace that you wouldn't be able to enjoy the experience as much, but it's doable for sure.
You will likely not encounter many people on the trail during the week but you'll see some. Bottom line: I wouldn't hesitate to ride the trail by myself.
There are a couple of people here that live near the trail and ride it a lot, hopefully they will respond too.
I've posted about my two rides on the Katy on my blog, if you're interested:
2010 trip report: http://www.loveofbikes.com/?page_id=942
2011 trip reports: http://www.loveofbikes.com/?page_id=1524
Good luck and you'll love the KATY, it's a wonderful trail!
Susan