You are correct, it is the score of the candidate protein against the BUSCO profile hmm assessed by hmmer. If it passes a threshold, it is validated as being that BUSCO gene. The threshold is defined for each profile based on the score of the species used to produce the profile. It can be found in name_odb10/scores_cutoff. And yes, the higher the better.
Most likely your Augustus installation is not functioning properly. Did you install Augustus from source or using conda? I believe the conda version of Augustus is known to have some issues.
Something worth trying: Running a complete busco install in a VM. This image has all dependencies installed and properly configured.
It's hard to say for sure without seeing the exact commands you used. However, Orthofinder identifies/defines single-copy orthologues as those genes present in a single copy in all species being compared in the analysis. If a BUSCO orthologue is present in more than one copy in one of the species being compared (could be multi-copy, could be fragmented etc.),...