The reason that this doesn't work is that your desktop isn't actually in vlan 200. You changed the IP address of the desktop to be on the 192.168.200.0/24 subnet but that's at layer 3.
Looking at the layer 2 topology, your PC is connected to a dumb switch which is hooked into a vlan 1 port on the managed switch. Because of this, regardless of what your IP address is the traffic coming from your desktop is hitting the switch on vlan 1.
Going back to layer 3 a bit, your default gateway 192.168.200.254 isn't in vlan 1 it's in vlan 200 because it's assigned to a router sub-interface which is watching for traffic tagged vlan 200. Therefore the PC's default gateway is unreachable at layer 2 and you get the infamous "Destination host unreachable" error.
For you to add a host in vlan 200, you need to plug it into a totally new port on the Dell managed switch and assign that port to vlan 200, as well as giving it an IP address in the 192.168.200.0/24 subnet.
On a related note, if you want traffic from vlan 200 to reach the outside internet you will need to create static routes on both the virgin media router and the Cisco router telling them about the other networks.
On the Cisco router you need to add ip route 0.0.0.0 0.0.0.0 192.168.1.1 to tell it to forward any traffic with a destination network it doesn't know about to virgin media. On the virgin media router you need the equivalent of ip route 192.168.200.0 255.255.255.0 192.168.1.254 so that the virgin media router knows where to send traffic coming back from the internet destined for vlan 200.
Not all together happy with how this post turned out, but I've learned things that I didn't know before, so that's good.
For example, normally if two nodes (A and B) are connected to a switch which is on the downstream side of another device like your firewall, the switch will route packets from A -> B and vise versa without contacting anything upstream (from @Tedwin).
I decided to post my own answer with the good parts from the comments with @Panther Modern.
Use higher quality hardware. Since I'm using Netgear ProSafe unmanaged switches, I'm considering upgrading to something like Cisco. @Mike Pennington suggested the SG300 series, which I have researched before and found on Amazon for less than $200. @Panther Modern suggests Arista, Cisco, Juniper and Brocade switches.
Troubleshoot each piece independently and isolate the problem. I've done some of this, to the extent that I have isolated the general path, but in an environment where 100% uptime makes people happy and allows them to get work done, this can be tricky. But point taken...
Consider topology and where you have access to and how many users you have. These factors can influence your choices for gear and how you go about answering your questions.
Find good networking tools and use those to help troubleshoot your issue. pfSense helped a lot concerning traffic that passed through the firewall. I also learned about Robocopy for limiting traffic on the client. This tool works pretty well.
Best Answer
Sadly, no. The unmanaged switch doesn't have the ability to create or trunk VLANs (if it did, it would be a managed switch). It will treat all the PCs as if they were on the same VLAN.
If it were a mananged switch, you would configure the 3560 port as a trunk port. But your unmanaged switch will not understand VLAN tagging.