What is the most Unity-compatible way to import sprite animations?

It depends on the system you are using to import the sprites. If you're using a 3rd party tool like 2D Toolkit, it easily creates sprite animations from importing separate individual sprite images with a consistent naming convention (i.e. walk_01, walk_02, etc). If you are using the built-in Unity sprite, it is easier to export the sprite into a single sprite sheet image, then import it as a new asset. Select the asset, then set its Texture Type to Sprite, and set the Sprite Mode to Multiple. Click Sprite Editor to open the editor. Then select the "slice" tool in the top left corner, and set it to "Grid" if your sprites are all the same size. If not, set it to Automatic. If you want full control, set it to manual. Click "Slice" then "Apply" and you're ready to use the sprites.

Next, select the small arrow next to the sprite in the Assets folder which will open up all of the individual sprites. If you have multiple animations within the sheet, you can select whichever sprites comprise the animation (or all of them) and drag them to the Hierarchy. Unity will prompt to create the animation. If you want just a single sprite, you can select one and drag it to the Hierarchy.

I would just decide which tool you want to use (check the asset store out though, there's a lot of good 2D sprite tools that will save you time!). Then export accordingly.