I can't really answer for the materials, as I haven't been able to export them myself. I had to recreate them myself in Godot. An alternative is to use the Godot Blender exporter, a plugin that exports the materials quite well. In return, I couldn't export animations correctly with it...
As for the animations. You showed the checkboxes that you ticked for the export of your model. There's an "Export Animation" one, that you need to tick so that the animations are exported.
I've had some problems with exporting animations, but I have found a workflow that globally works quite well in most cases. Firstly, do all your animations with an Armature. It means that you won't be able to animate the modifiers of the Mesh, unfortunately. Armature will be a bit cumbursome, but it still offers a lot of possibilities, so in most cases it's sufficient.
Then, once you've done your animations with a single armature, make all the meshes that are animated a child of the armature. If you don't, the Better Collada exporter will curse at you and all the exported animations will be wonky.
Once you're done with that, go to the exporter and tick everything except "Selected Objects", "Apply modifier" (except if you need to), "Tangent Arrays" (not necessary, but do if you need to) and "Shape Keys".
Normally with that workflow you should have the same animations in Blender and Godot :)
Hope this helps