Is a haploid essentially half of a homologous pair? I understand that a haploid consists of only one set of each chromosome, so does that mean that if that chromosome was replicated the cell would become diploid and have homologous pairs?
Yes, you seem to have understood it correctly. On top of haploid and diploids, there are also cases of triploids, hexaploids, octoploids and others.
For example, some species of strawberries are decaploid (10 copies) (Hummer 2012). Then, there are the extremes. In Entamoeba populations, there can be variation in ploidy level among individuals ranging from diploid (2 copies) to tetracontaploid (40 copies) (Mukherjee et al. 2008)!