Yes (kind of). You can upload svg files into HyperTheme but scaling needs to be fixed at the moment. Currently the image becomes pixelated when enlarged but this has been reported as a bug and will be fixed.
You can upload a bunch of file types (tiff, bmp, jpg, etc), but as dragon57 said, png is usually the best for HT.
Believe it or not, the online photopea editor seems to have full PSD support: https://www.photopea.com
A number of other Photoshop alternatives exist. I personally prefer Affinity Photo, and it imports PSD.
As to what filetypes, you really can't go wrong with PNG, which has transparency support, which is needed for lots of effects. I'm sure others will respond with their favorites and add more to this. Best of luck.