A qualified electrician was either of two types, skilled or competent, but latter was dropped, but skilled means "A person with technical knowledge or sufficient experience to enable him/her to avoid dangers which electricity may create." it does not mean he has a City & Guilds certificate or apprenticeship deeds. So yes the work should be done by a skilled person, but it is possible that for that work the landlord is considered skilled.
There is a scheme member electrician, which means they pay an annual fee to be able to do some work and inform the LABC after complete rather than before it is started and it costs them less for each notification. That does not make them any better than an industrial electrician in fact likely less skilled, but it does allow them to self certify.
Any electrical work should include inspecting and testing and the issue of a minor works or installation certificate. However it is only I think required to give tenants a copy of the electrical installation condition report (EICR) every 5 years, don't think they have the right to view the other paperwork, but not sure on that, a new law has come in, and where the installation certificate is under 5 years old an EICR is not required, so maybe he does need to be provided with copies of installation certificates and minor works certificates now?
It was forbidden for electricians to pass on the certificates to the tenant, it had to be given to owner, who in turn could give a copy to tenant, but not sure if that has changed.