Also, can one check if keystrokes are being recorded?
If the firmware on your router is replaced by a malicious program, then it could intercept all SSL connections and handle them itself. However, it will not be able to present a valid certificate, signed by a real Certificate Authority. Thus the lock icon will show as broken, and you'll know that something is wrong.
Note that in a corporate setting, it's not unheard of for an employer to add itself as a trusted Certificate Authority on your work computer. This allows them to intercept all encrypted SSL traffic and also have the lock icon not be broken. Of course there's absolutely nothing wrong with this, because your employer is perfectly trustworthy and would never betray you or spy on you.