no that is a crossover adapter, it allows PCs to network directly with only a cable.
pc>cable>pc. most modern NIC Network Interface Cards can auto detect and crossover internally so adapters like this are mainly for older equipment.
the wiring in the wall should be able to carry data, but you may get tons of crosstalk if the wires are not TP - Twisted Pairs.
you would need to put a network punch-down block where all the phone lines converge, change the faceplates in each room to be RJ-45 from the RJ-11. once the faceplates are changed and the block is tied into the router you should be good to go.
there are other steps I missed but as you can see its a weekend of work, plus tools and equipment. maybe a router with MIMO and a better wireless NIC are in order?