Not sure you fully understand the real purpose of a router.
Not only is it to make having a LAN easier, with the built in DHCP server assigning IPs to your hardware on your LAN, but to separate the public, internet, side from the private, LAN, side. Blocking all incoming traffic from the public side with NAT (Network address translation), which is basically a rudimentary firewall, unless YOU SPECIFICALLY allow that traffic to enter your LAN by opening and forwarding ports to a specific IP address inside your LAN.
None of this has anything to do with WiFi. WiFi is on your LAN and gets the same security from the public, internet, side as does all the hardware wired to the router.
Where security is needed on WiFi is to block people, Computers, from connecting to the WiFi signal without you specifically allowing them to connect.
No one from the internet side can logon to your WiFi signal.
MAC filtering is not security. No need to use it, IMHO and many others. Disabling SSID is no security. It only makes it harder for YOU to connect to YOUR WiFi.
Ping Sming. Don't worry about it. So your router is pingable. Ping IS PING. That is all it does. My router is pingable also. Has any of my PCs inside my LAN ever been hacked through ping or any other method, NO. And I have a static public IP that is hit all the time. I can post the logs from my router to prove that. I also have ports open so "I" can get to PCs on my LAN from the outside, internet. I use Remote Desktop to connect to my home LAN and for the longest time I used the default port 3389. Even with that I have never been hacked because I have a very secure password on all my PCs. So even if someone tried to connect to RD, and people have tried, they would be met with a user name they didn't know and a password that would take forever to figure out, including the password on the built in Administrators account.