do we still need a static ip internet service if all we're doing is setting up a NAS?
There is no fundamental difference between a NAS and a server. A NAS is just a server in a small housing. So yes, you still need a static ip, and no, you don't.
The whole point of having a static ip is to know how to reach your server from outside. A static ip solves that problem by offering a fixed address. But there is also a solution for dynamic ip's, called DynDNS. A service (which preferably runs on your router, but it can also run on your NAS) keeps track on your ip, and as soon it changes, it's passed on to an external dns server, so you can reach to your network at something like <youraccount>.dyndns.com.
My second question would be should we connect the NAS directly to the router or to the the switch which is connected to the router?
Doesn't matter, as long as it is at the LAN side. (So not between the router and modem, the WAN side).
So if I connect a NAS to the switch, will it be accessible remotely? Will it have a static ip?
Maybe. You'll need to forward some ports in the router, to enable the services you want. (A NAS can run a whole bunch of services, all needing their own port(s))