If you're buying in a power supply (you don't indicate open frame or wall wart so I'll assume wall wart), then you will at least want copies of their certifications for your Technical Construction File, and have sufficient confidence in their validity, ie. that
they haven't self certified. They should have used a reputable test house, particularly for LVD (safety) testing. You can't just switch PSUs on the fly either, as that would affect your overall certification. If it is any sort of open frame PSU then you become responsible for LVD too, its safe enclosing, components, wiring etc.
With LVD taken care of, then yes, you have EMC to worry about. You need to purchase the relevant standard for the equipment that you are building (there are no 'freeware' copies) and verify compliance. On some equipment, you can do the sums and prove that you can't possibly emit sufficient RF energy to exceed the limits, but that certainly won't be the case for a mains block powered device.
You really need to do radiated and conducted emissions, using your chosen mains block. You may also need to do susceptibility and ESD too, depending on what the function the device is and the consequences of it failing. (ESD, you can probably do yourself to a reasonable level of confidence - you probably want to do this anyway if you want to be confident about not getting lots of customer returns).
You didn't say if you possess any EMC pre-compliance test equipment, but without it, I can't see that you would have the evidence needed to attempt self-certification.
P.S. Just to try to add a few specific answers to your questions:
1) There are laid out in the specific standards, based on product type. At the moment, UK and EU standards are still harmonized (according to the BS-EN numbers on the BSI site anyway). US is based on FCC standards. For example, you can find the UK EMC standards at
https://shop.bsigroup.com/Browse-by-Sector/Manufacturing/Electromagnetic-compatibility/Electromagnetic_compatibility_standards/.
2) Yes you can self certify, given
sufficient evidence that you comply with the relevant standards.
3) The method of self-certifying is by creating and maintaining a Technical Construction File (TCF), which documents the
evidence that supports your claim to comply with the relevant standards. This must be available for examination in case of a challenge.
Basically, the test / compliance requirements for your (quite specific) product type comprise EMC (Emmissions - Radiated and Conducted, Susceptibility - Radiated and Conducted, and ESD) and LVD (Low Voltage Directive - Safety). Talking to a friendly local test house can save you an awful lot of time - Consultants (of any description) tend to be expensive!