In short, being haploid or polyploid should not protect you from Selfish Genetic Elements (SGES).
Almost all eukaryotes are polyploid. Plants are the oft-cited example. Wheat, I believe, is hexaploid for example. Polyploid bacteria are the exceptions typically (if we consider ploidy to be of the whole genome). However, even in truly haploid bacteria, the genome in any given cell is usually actively being replicated, so some of the genes are actually present more than once.
Regardless, replication and repair is an enormously well controlled and regulated process: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1479556/
SGEs can take many forms depending on your definition and are ubiquitous in nature as far as we can tell. 667
Plasmids which harbour addiction factors, insertion elements, transposons, prophages etc are all technically SGEs. See here for a review: https://www.ncbi.nlm.nih.gov/pubmed/21227262
I can't find any literature on SGEs in polyploidy directly, but it is well documented that bacteria harbour a considerable amount of selfish DNA - it seems likely that this will be the same if not worse in polyploidy. Plasmid detection is a difficult computational and experimental process (http://bioinformatics.oxfordjournals.org/content/early/2016/11/18/bioinformatics.btw651.full) but one of the hallmarks of plasmid sequences are typically addiction factors (a gene that the host cell must maintain in order to survive - usually an anti-toxin to neutralise a toxin present on a different DNA molecule) as well as phage genes.
To address a couple of things specifically:
As each copy replicates separately it would seem likely that spontaneous mutations would arise and over time
This is somewhat true, there is always minor variation in a population (else evolution wouldnt work!). Sometimes you'll find this is also the case in a single cell - but less often. All cells we have found so far have DNA replication and repair mechanisms - https://en.wikipedia.org/wiki/DNA_repair. The frequency of mutation in the genome is not directly linked to the amount of selfish elements though, making your second point contentious:
This seems like an ideal place for selfish copies to arise
I agree with the premise of polyploidy probably inviting more SGEs (thinking specifically of IS elements and transposons here really), but not because of mutation - rather, just because there are more DNA molecules and sites present for those elements to incorporate into. 667