GCN Network security an configuration issues
Solution 1:
On a rented room we bump into a new device. It's a sim card router, so there is no telephone line (DSL) or fiber cable
Sounds like an LTE (4G) modem.
Apparently this device uses CGN technology and for practical purposes it only means that the router doesn't have a public IP and is instead embedded in a broader private network.
CGNAT is used by the network operator, not by the device itself. Your router only implements NAT for your own LAN, the same as with other home routers, but is mostly unaware of CGNAT happening.
Most 4G operators use CGNAT, although some occassionally offer a "static IP address" as an additional service. (Many DSL and fiber ISPs have started using CGNAT out of necessity as well.)
Is it possible to other networks on the second layer to sniff our network traffic?
This depends mostly on the network technology. With LTE it should not be possible, as each device's data (be it a cellphone or a fixed modem) is encrypted using a different key.
The usage of CGNAT doesn't affect this at all.
Being that we are on a LAN inside a LAN would there be a problem if the gateway is the same IP number than the "WAN"? This is to say, I configure my LAN with 10.XXX...
Yes, it could – however, just being "10.XXX" does not make it the "same IP number". Networks stopped working that way several decades ago. Now 10.xxx is only the reserved range, but it can have several completely independent smaller networks (e.g. 10.0.x.y, 10.1.x.y, 10.2.3.z ...) which wouldn't conflict.
In other words, whether it's actually "same IP number" depends on the prefix length (i.e. subnet mask) configured for both sides. For example, if your LAN had 10.0.0.0/8 (i.e. 10.x.y.z) this would conflict with a WAN address that is 10.1.2.3 – but if your LAN had 10.0.0.0/16 or 10.0.0.0/24 (i.e. 10.0.x.y or 10.0.0.x), it would not conflict.
(As far as I know, LTE uses point-to-point links, so if the modem doesn't show the netmask or prefix length for its WAN address, then it can be considered as /32.)