I was taking the burst photos and really liked how they looked when I scrolled through them quickly. Is there an app that will let me combine images into a gif-type file or video so that I could save them to the computer or put them on Facebook (without having to give the app my Facebook login)?
If you used the Microsoft Blink app, then you can export .gif photos to your OneDrive* account from within the app itself. But if you used a different app, e.g. the Microsoft Camera app, then you might need to save all of those burst images to your PC and create the .gif there - browsed through some .gif apps in the store just a few moments ago and it seems like none of them can create .gif images from images already saved in the phone.
Regarding Facebook, there seems to be no way to "put them on Facebook" without logging in from the app - you will be required to log into your Facebook account before you would be able to post anything into it.
*- Once in OneDrive, you should be able to view it in your computer either through the OneDrive app/software or through your web browser. Additionally, the Blink app also has the option to share via e-mail ; email to yourself and then download the attachment.