Multiple actions are supported in glTF-Blender-IO, I’ve checked this quite recently… they do need to be either active actions, pushed down as NLA tracks, or stashed actions.
It is pretty much impossible for us to diagnose the texture issues without knowing what sort of Blender material has been used. I’d suggest Principled BSDF if you’re using Blender 2.80. In any case, if the model doesn’t work in multiple glTF viewers then you’ll have much better luck filing a bug on the exporter(s) than in the threejs forum.