I do these systems all the time and it's interesting to see the different approaches to achieve the customer requirement.
this is how I would do it.
Route a cat5e cable from each room where a hard wired internet connection is specified back to the main cupboard or termination point, ideally where the other services in the home are located.
Route a cat5e cable to where each of the 3 wireless access points will be located, these are best located fixed to the ceiling and some very discrete access points are available that blend in nicely but give great coverage.
this will mean you end up with a bunch of cat5e cables in one location star wired to each room and the 3 access points.
the customer will provide the modem/router and you will either need to terminate the cat5e into a patched panel or just stick an rj45 connector on the end. Either way each cable will need to be connects to a switch and these are inexpensive. A 24 way switch should suffice in this case. You then connect one Ethernet cable from the router to the switch and this produces the local area network...it's easy if you avoid all the terms involved as it guys love to abbreviate stuff and make it sound more difficult than it is!
Terminate the other end of each cable into an rj45 faceplate or module and stick an rj45 connector where the 3 access points will be located.
from experience you are taking a massive gamble if you use wifi range extenders as they are subject to a single point of failure and can be very unstable/prone to interference. If you hard wire an access point it will be very stable and give you no issues.
are you configuring the access points too?
hope I've helped but feel free to ask anything you need, if you need a part list and diagram just give me a nudge.