You don’t actually have to use the server that the certificate gets installed on to generate the CSR and first install it, but it is generally a good idea to do so just so you can get the certificate installed into the right place right away.
If you have access to the server where you need to install the certificate you might as well use it.
Depending on the kind of certificate you purchase this name will show up on your certificate. those that cost under $100 for single domains) don’t list the organization, the higher signature certificates that also require extensive validation by the cert authority do.
Regardless you should make sure this matches the right company/organization. Not really sure what this is for, but traditionally I’ve always set this to Web because – well this is a Web thing after all right?
When it works it’s fine, but when it doesn’t man can it be a pain.