Should SNPs that have this sort of name 'exm_...." be removed from genetic data at the QC stage.
Not necessarily, they used this ID cause it was part of their ExomeSNP array, probably because there was no RSID at the time, for example this one:
Should SNPs with alleles codes such as 0 and A be removed giving one is missing?
Yes, they are probably not SNPs, so it should be safe to remove them.
Should SNPs that have a kgp or JHU prefix be removed? What are they?
Here is how you can convert them to rsIDS:
There should be other methods as well, like annotating with DBsnp, I wouldn't worry too much about the IDs as long the quality of the data looks ok.
Should chromosome 26 be removed?
Yes you can remove, here is anther post about them:
QC of genetic data
What about SNPs that have a SNP name such as 1 seq 0 12002028. A. G
Probably SNPs that still don't have an rsid, like in the first example.