Can a Full Scope Domestic Installer (with Part P) inspect and test other electricians work?

No problem at all with later inspecting and testing any circuit as long as your only completing a EICR form. Electrician A's problem is that you can't use a minor works cert for the installation of a new circuit, this would be notifyable under building regulations, nor could he install it and you do a EICR form for it... if it is only an alteration or amendment of a circuit then Electrician A can do a Minor works cert and it wouldn't need signing off, anyone could then carry out an inspection and test and provide an EICR. If what you actually mean is can they install it and you sign it off for building regs then basically your options are that they notifiy local building control before carrying out the works, do a 'green cert' for the installation and have Building control sign it off (at a cost). Or the install and you certify on a 3 person certificate, electrician A signs as Designer and Constructor, you sign as Inspector... You are signing off and it is under your registration if there is anything wrong so don't just put your autograph to anything!

i think cann't issue EIC on other's work. But EICR can do

Who are you registered with? You can do an EICR to any existing circuit, but a minor works does not cover a new circuit, or major alteration to an existing one as you are changing the characteristics of said circuit. To issue an EIC as a 3rd party: To my knowledge, only Napit allow you to do this, however I can test other persons work but you have to be involved at all stages and have the necessary experience. Once you have issued the EIC, a panel will look at the certificate and review the test results etc before it then gets sent to the LA. I think you may be assessed on this annually too, but not sure. If in doubt, ring your scheme, that is what membership is for. Personally I do not like to sign someone else's work off. Have you been involved with the build from the beginning and seen the work at every stage? Your name on the certificate, your liability!!