Is it expected that I use the previous assembly's annotation file along with the new assembly's fna file?

More than likely not in this case. If you look at the three assemblies, they are using different sequencing technologies so more than likely the genome coordinates are not going to be identical.

Also, what does it really mean that the new assembly is not a RefSeq accession but is a Genbank accession?

Only one of the assemblies is designated as RefSeq. It has likely had more curation than the other two but that may have been just NCBI's eukaryotic annotation automated pipeline run. At some point this newer assembly may go through that process but for now it has not.