As long as the NAS and your computer are all on the same subnet, you shouldn't ever be touching the router. Is your computer also getting its IP address from the router? According to your diagram, there doesn't appear to be anything on a different network, subnet or VLAN, so I'm assuming there's only the one DHCP server.
You can verify the route your packets are taking by running traceroute (tracert on Windows) from your computer to your NAS. The path that traceroute prints will show you exactly how you're connecting. In your case it should just print your computer name and then the NAS name/IP.
For example, if I traceroute my file server at home from my desktop this is what I get:
$ traceroute thoth
traceroute to thoth (192.168.2.14), 30 hops max, 60 byte packets
1 thoth.reeds.local (192.168.2.14) 0.421 ms 0.417 ms 0.403 ms
Which shows no additional hops even though it passes through a switch. Conversely, if I traceroute to something on my lab subnet from my desktop, you see it pass through my router even though it's physically in the same location:
$ traceroute 172.24.20.21
traceroute to 172.24.20.21 (172.24.20.21), 30 hops max, 60 byte packets
1 set.reeds.local (192.168.2.1) 1.345 ms 1.313 ms 1.268 ms
2 172.24.20.21 (172.24.20.21) 1.891 ms 1.536 ms 1.548 ms
If you're seeing your router's IP in there, something about the way your NAS or your computer is setup doesn't make logical sense. There's no reason to be subnetting, or using VLANs on a network as simple as yours, in a home situation.
Are you able to transfer files between two different clients attached to the same switch as your computer at gigabit speeds? What about with two clients plugged into the switch that your NAS is?
You can test your NAS read speeds by doing:
dd if=/some/big/file of=/dev/null
This takes /some/big/file and reads it from disk and writes it to nowhere. It will give you a basic (but not perfect) idea of your sustained read speeds. In dd, if=... means InputFile, of=... means OutputFile.
You can test your NAS write speeds by doing:
dd if=/dev/zero of=/somewhere/test.file bs=9000K count=1000
This will write a 9GB file filled with zeros, you need to make sure that the file you're writing is larger than your write cache (if you have one on your NAS) because otherwise you'll see something ridiculous like 48GB/sec