I'm wondering what could a hacker do if he had access to my router?
An attacker (the hacker) could do anything if Telnet or SSH access is available (probable). If the router's shell environment allows writes, then the attacker could run
echo "binary contents of executable">executable;./executable which could be used to execute any executable. The attacker could do anything that the modem does using software (intercept, modify content received or sent).
Even without Telnet or SSH access, the attacker could use the web admin utilities to change DNS entries and redirect any website to a malicious one.
In any case, communications over HTTPS (including passwords) will not be compromised as the data is encrypted before reaching the router.
My router does not show my external IP.
All routers must have an IP as they send IP packets which have an IP address and destination.
PS: It's same situation here in India across multiple service providers.