Netgear Modem CM1000 v2: How many devices can I connect?

> I own the Netgear CM1000 v2 modem that I have connected to a netgear
> switch via Cat 7 Lan cable. [...]

   Not a good plan.  (And "a netgear switch" is not a very detailed
description of that device.)

> Wired: Modem--switch-->devices (not working: can connect the router
> and a laptop but the modem doesn't 'see' other devices)

   Yup.  Less vaguely, additional devices are not getting connected to
your ISP through the modem.

> Wireless: Modem-->switch-->Nighthawk 7000 router (works without any
> problems)

   Yup.  And you should be able to get the same results if you remove
the switch.  (Assuming that "Nighthawk 7000" means "R7000".  Look at the
product label.)

> I called Xfinity and they confirmed that they aren't placing any limits
> on the number of connected devices on my modem.

   Not strictly true.  They limit you to one device (one external IP
address).  That one device is normally a NAT router (like, say, an
R7000) which would allow your one external IP address to serve multiple
devices.  Whoever you talked to probably assumed that you had a router
connected to your modem, and they don't limit the devices attached to
your router.

> How can I resolve this problem?

   Connect your modem to a router (WAN/Internet port), and connect your
client devices to that router.  If you have too many wired devices for
the LAN ports on the router, then you can connect a network switch to a
router LAN port, and then connect more devices to that switch.