I too, have been searching for a similar app, but just for myself. I use Outlook at work, & like the scheduling part, but have not found any small app that does similar, until recently.
I have been using Vorg Express Version 1.6 something from www.vertikalsystems.com.
In addition to the standard daily, weekly, monthly, etc., it allows recurring patterns such as the 3rd Monday of the month. But other than that, I don't think you can mix patterns.
Like Outlook, you can have multiple resources (people) for tasks. If you want to categorize items, you can add new categories, & add your own icons to them.
I also like how I can minimize it to the System Tray instead of to the task bar.
The icon does have a little flasing red square on part of it, when a reminder has been activated.
CONS: I don't like the way it starts up in full screen, instead of remembering the state it was in when it was closed.
The pop-up reminders are not persistent, and & come up fairly small, & from the systray. Quite often I do not notice them sliding up, & therefore miss the reminder, as I normally keep the taskbar hidden. I have asked for this feature to be updated, and they are looking at that, but no word on how soon it will be added.
There are times that I think the app is a resource hog, & slows down my old system when left running in the background, though.
There is also a commercial version named VORG, which has more features, I presume.
Email tech support is fairly responsive, usually within a couple of days at most.
Since the Vorg Express version is free, you might give it a try.