What is the error rate for these machines in genotyping (like for 23&me)? What is the chance of a false positive or false negative?
23andme uses DNA microarrays ("chips") to perform genotyping. These methods only assay a set of previously known single nucleotide variants, so the probability of missing some specific variant (false negative) is high if that variant is not specifically assayed. It is not quite 100% as frequently known variants missing from the chip are closely associated with a variant that is on the chip, so you can sometimes impute missing data.
Sequencing-based methods will in theory pick up unanticipated variants, but they have their own foibles.
Generally both methods are quite good at single nucleotide variants (SNVs) but are bad to only ok at other kinds of variants. See for example (this paper (chosen somewhat at random, there are many similar papers) to see what the state of the art is for sequencing-based non-SNV calls.
Getting back to chips, these themselves depend somewhat on what version of the genotyping chip is used. For specific information on different versions of 23andme chips, I recommend this blog. For tech specs on the most recent Illumina chip used by 23andme, you can see their product sheet, and specifically Table 5. Their data claims that call rate among targeted variants is >99%, and among those the reproducibility is >99.9%. However things can be reproducibly wrong, so that's not exactly false positive that you requested.
Though it's somewhat old, this paper will give you a sense of both technologies in terms of variant ascertainment, claiming 89% sensitivity and 99.6% specificity (for European genotypes only) for either low coverage sequencing or a then-current chip.
To learn more about how these stats work for chips, specifically for false positives, this video may help.
Hope that helps.