Does the Houdini Engine API support saving bgeo files?

I want to create a digital asset that will the user to add attributes to a object and save it as a bgeo. Is that possible?

Does the matra node work in the Engine? I want to be able to render images, using Houdini's Manta renderer.

Yep, you can bake out bgeos with the HAPI_GetGeoSize() and HAPI_SaveGeoToMemory().

Mantra rendering should also work. You just need to initiate a render via an exposed button (or other parameter) on your asset. Houdini Engine does not expose direct control of Mantra.