Use my Apple ID on another device to download a previous purchase

Yes

Open App Store app > Updates > Purchased

Yes, you can do that, however, you should only do that on your own iOS devices. You cannot share your iOS content with anyone but immediate family and you actually shouldn't sign into your Apple ID on someone else's iOS device or they will be stuck with only your Apple ID for the next 90 days.

Let me see if I understand this correctly..... My wife and I have separate apple IDs. And she wants an app that I have purchased. I can download the app she wants on her iPad using my ID. But if I do this she will be unable to use her ID on her iPad for 90 days?

That's what Apple states;

Your Apple ID can have up to 10 devices and computers (combined) associated with it. Each computer must also be authorized using the same Apple ID. Once a device or computer is associated with your Apple ID, you cannot associate that device or computer with another Apple ID for 90 days. You can view which devices or computers are currently associated, remove unused devices or computers, and see how long before they can be associated with a different Apple ID from the Account Information page in iTunes on your computer:

http://support.apple.com/kb/HT4627

Ok thanks