Do I need a router with Mac and these firewall settings? I am single home user in one apartment with no other users, living apartment block house with over hundred apartments.
I have cable internet and one modem.

Mac firewall has "block all incoming connections" and "enable stealth mode" enabled.
All sharing services are disabled.

Do I need a router? All ports are disabled with these settings (I assume at least) so is router needed?
You only need a router if you want to "route" a network connection to more than 1 device. 

If you only want to have 1 device connected to your network then you don't need a router. If you add another network-enabled device (TV, printer, another computer, etc), then you will need a router to share &amp; control the network traffic.