Not quite. What most people refer to as their modem is actually their router/firewall. It can be your modem too depending on how you get your internet, a modem switches your connection from one physical delivery to another such as cable to ethernet. What most people refer to as their router is really just a wireless access point with a few ethernet ports. What ai think you are referring to for why it's good to sit behind a router is the NAT(net address translation) which takes your public IP and converts it to a private IP range usually in the 192.178.0.0 range. You wouldn't be able to do anything if you tried to directly connect your device to your public IP since it probably wouldn't know the proper routing protocol to talk to the upstream device. Basically your fine plugging into whatever is provided to you by the ISP.
yeah it's awful. Orignal PS4 doesn't even support 5ghz, only 2.4.
