virgosystems

Is sshd running on your Pi? What do its logs say?
 
Does your network have a route from 192.168.137.3 to 192.168.1.10? Can you ping it?

This sounds quite confusing.
 
PHY is set to DHCP by default so it tries to aquire a lease by a DHCP server available in the network. If you don't have one you have to find another way to make the board connectable via copper. Like using UART serial console or build your own script using custom configuration.
Cannot say about using Pi as DHCP since I dont know your full network setup and therefore no idea if that would work. A dhcp server is usually also some kind a router to NAT data to a different network or WAN.

Close the session or use sudo -k which forces a time out.
More info in this link https://askubuntu.com/questions/1195719/how-do-i-force-sudo-to-ask-for-a-password-each-time-when-a-specific-command-is

Sudo users by default supply their own password for authentication, rather than the password of the target user. After authentication, and if the configuration file (typically /etc/sudoers) permits the user access, the system invokes the requested command.  https://en.wikipedia.org/wiki/Sudo.

When you ran the command 'sudo passwd', you were changing the password for the root user, not the user invoking sudo (you were acting as root when you changed the password).