You need a router for Internet connection, yes. The EnGenius EAP-3660's are not routers, just APs. They also support WDS bridging/repeating without flashing with Open-Mesh. WDS would be a bit more complicated to set up, but would provide similar throughput to a mesh setup.
Using 802.11g gear, you should expect throughput in the low single digit Mbps (1-5 Mbps). Might be ok for a low bandwidth video stream like YouTube or other Internet video, but hard to tell.
If you can't run Ethernet to any of the APs, then the router you choose will also need to support WDS. But then it can't be used with the mesh APs.
It's bad that your Ethernet location is in one corner of the building. That means that the far-end room will have the worst throughput because wirelessly connecting it will require multiple "hops". (Each wireless AP that a signal has to go through before it hits a wire is a "hop". Each hop costs you > 50% of bandwidth because the radio has to receive, then retransmit.
Best case throughput with strongest signal for 11g will give you ~ 20 Mbps throughput. But more typical signal strengths will start you off with more like
10 Mbps. You can do the math from there to figure out the throughput you'll get at the far end.
Best to try to get your wired connection to the central room if at all possible.
I know it's a stone church, but how about crawl space, heating ducts?
