Router cannot connect to Internet
I would guess that the problem is due to both of the router's interfaces (both WAN & LAN) having overlapping or even identical network numbering.
You say that the router picked up 192.168.1.1 as the upstream WAN gateway (the network is likely 192.168.1.0/24), and your DHCP server logs show that it's offering 192.168.1.x addresses as well (the LAN network is again likely to be 192.168.1.0/24). Some routers are able to cope with this situation, but nevertheless it's a situation I'd run away from.
So if you want to use it in full 'router' mode, try changing your router's LAN configuration to use addresses from e.g. 192.168.2.x or such. (It can be any private address range as long as it doesn't overlap with the "WAN" addressing.)
(You could also use the device in 'bridge' mode; instead of creating your own separate LAN, it'd directly place all your devices on the owner's main LAN. Though in this particular case I'd be hesitant to switch to bridge mode due to security/privacy concerns, but if it were a 2nd router for your own WAN, bridging would be the preferred option.)
The date is Dec 31 because the router has no better clock information yet – without Internet access it cannot synchronize from NTP servers; it likely has no battery-powered clock to retain time from previous time; and you haven't set the clock manually yet either.
Based on your logs and post, I posit that the issue is that both routers are using the same 192.168.1.0/24 (or possibly bigger range) for IP addresses, thus computers on the LAN of the router that is not working can't route correctly.
The easiest fix might be to change this routers config so that the LAN is on 10.0.0.0 (netmask 255.255.255.0 will work fine), ie set the routers LAN IP address to 10.0.0.1.