Now, for me, as an attacker, is it possible to use (any) IP (just assume ip pool is 192.168.1.x is allowed), which the LAN router wouldn't detect and log in its general logs, (like DHCP) leases and so on?
It depends on the router. Cheap SOHO routers, once programmed for a given DHCP address pool, will blithely accept packets from addresses in the pool even if they did not give out those addresses.
In such a scenario, just grab IP address 192.168.1.201 and Bob's your uncle.
That very same strategy would lead to an alert being immediately logged by a more clever router as an obvious anomaly. Often, static IP alerting is set to "off" in order to accommodate users which may want to assign IPs in a quick-and-dirty way (e.g. printers) and don't want to be bothered with properly partitioning and configuring the network.
So basically it all boils down to the network "budget" in money and care. A carelessly configured or underbudget network will let you effortlessly connect unlogged, and actually it's likely not to have logs in the first place, or maybe there are but nobody remembers them anymore (or they were received by someone that long since has classified them "spam": I've seen that happen).
On a more carefully maintained network, maybe with a higher-end router with features that are known and used by the network administrator, you probably won't even be able to connect because your MAC address is not known. You may be able to circumvent that, if you change your MAC address to that of a device that's known to the system but not currently in use (e.g. a printer - and they often have the MAC printed on the network plaque on the back, or it's on an easily reachable "Printer Status" page that can be either displayed or printed). Even so, you'd be wise to plug the computer in the same port where the printer was.
(I seem to remember a blog article somewhere on the possibility of running
nmap on newly connected devices recognized through ARP querying, both for fingerprinting and security auditing. I can't find it just now. But in theory at least, even if you connect to an unencrypted wired LAN with a valid MAC and from that MAC's port, it is still possible to detect that you aren't a printer after all).