How could cas9 be inserted into cells by researchers looking to edit a genome? I imagine for engineering bacterial systems you could just put in the cas9 coding region in an expression vector, but is that how it is also done for eukaryotic cells?
If you are talking about the nuclease functionality of Cas9, then you aren't adding the Cas9 gene into the genome, you are transfecting it on a plasmid. Usually there is either some form of permeabilization done to the cells, or the construct is inserted into a viral vector in order to transfect the cells in tissue culture. In multicellular animals, you would need to use the viral vector route as would need to be done with any gene therapy technique. Addgene is a nonprofit plasmid repository that has many of the CRISPR/Cas9 constructs that have been developed, and they have an excellent resource on their Website.
You wouldn't really want Cas9 active in the cell after it has done its job, as you increase your likelihood of off-target hits. The target shouldn't be in the genome anyway if the targeting event occurred, because you have either substituted in another piece of DNA or you have cut the DNA and the DNA repair mechanisms have introduced random nucleotides in order to repair the double stranded break.
Eukaryotic cells, for the most part, cannot maintain plasmids. The Cas9 gene will be transcribed along with the guide RNA. The resulting protein/RNA complex targets the sequence of interest and makes a cut. When the cell replicates the plasmid will likely be lost, but the daughter cells will inherit the genomic edit made by the targeting event.
If you are looking to insert a different gene at the target site, say a reporter into the genome, then you co-transfect a linear piece of DNA with homology arms around the cut site so that that donor will be inserted via homologous break repair.
One of the Cas9 constructs that has bene developed has disabled the nuclease activity, so that the Cas9/RNA complex ends up acting as a binding protein to suppress the activity you are looking to study without actually deleting or replacing the gene. In this case you may look to insert the Cas9 cassette, in which case you would use a recombination event as you would with any other type of incorporation into the genome.