The existing extension wiring may or may not be suitable. Who installed it, and have you any idea what the specification is? Does it have anyything like CW1308 printed on the cable?
Is it hidden behind walls or just attached to skirting boards and such?
The engineer can in any case install a Data Extension Lead
from the master to your extension, (or elsewhere if you like), and put the modem at the current or new extension.
A decent engineer would convert the extension into a master, yes. Is the socket in the hall completely unnecessary, or would you need to be able to use it?
Yes, the modem, router and phone base station can all be connected into a 3-way power extension lead.
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