I'm working on an HTML Application for Windows 8.1, and, as our office soon will use Windows 10, I'm wondering if MS Edge and Windows 10 still support HTA.
No. However, you can still run legacy HTAs in IE9 mode. For new development using web technologies, Microsoft is recommending a switch to Windows Store Apps.
Here's what Microsoft had to say about HTA support in IE10 and later:
The Internet Explorer team is increasingly focused on standards compliance, and markup-based behaviors are not part of modern web standards. In IE10 mode support for markup based behaviors has been removed, and this includes hta:application.
In addition, for HTML-based applications the focus for Windows 8 and beyond the team's focus is on Windows Store applications.
I suspect that an HTA would work in Windows 10, but only in IE9 mode. (EDIT: It does!) It would be a major shift if Microsoft built Edge with HTA support.