Ok, let's go with facts here, not assumptions.
First of all, Cat5e cables are plenty, you don't need Cat6 unless you're running 100m+ lengths of it.
The QNAP TS-412 is a rather old and "slow" NAS by modern standards, but at the same time, you should see higher speeds than what you're getting. Normally you should see higher performance reading data from the NAS compared to writing data to it.
Are you sure it's 1-5Mbps (Megabits) or is it 1-5 MB/s (Megabytes). This is a very important distinction to be made. If it's 1-5Mbps, it's super slow and something is very wrong. If it's 1-5MB/s it's still slow, but 5MB/s is about what the NAS can manage in terms of writing data to it. Read speeds should on the other hand be closer to 9-10MB/s.
A couple of things to check is that you 1. don't have any cable issues as mentioned 2. check that not one of your network cards (including the NAS) is set to operate at 10/100Mbps rather than Gigabit speed. 3. Check that the NAS isn't set to use Jumbo Frames when the PC is not, as that would cause some issues. 4. try taking the switch out of the calculation, test again and then do the same with the router and use the switch instead. If one or the other has an issue, you should spot it here.