First: Avoid the J-series Synology units. All J-series are meant to be cost-reduced, low power units, typically based on ARM processors. They will perform rather poorly, and be completely useless for any kind of video editing applications.
So when I get a NAS, what can I expect?
Lower performance than USB 3, but greater convenience because all your devices can access it simultaneously. You're going to want your computer and the NAS to be on the same Ethernet switch. Don't even think about doing anything serious over Wifi.
Would it be a good idea?
Yes and no. Yes, in that it could make a great backup target, and a handy way of storing non-video content for live use, like music libraries.
How is performance when connected to a network?
Depends on the NAS and the network. As I just finished writing up over here the rest of the traffic on the network can play a role in limiting performance, and the functional capacity of the NAS itself also plays a very key role in whether or not it can keep up with what you demand out of it.
Is it totally doable for live editing or is it best connected directly to my PC via USB 3?
Maybe. Depends on the footage you're working with. Unless you're getting a higher end model, know how to tinker with these things, and you're not working with like gonzo 4K footage it might work out, but USB 3 will deliver far superior performance, and it would be recommended you continue to work off something like this.
I'm less concerned about RAID... I'm really only doing this so I can have multiple drives using one power cable.
No, you're getting involved with RAID, so you need to be concerned about RAID. I have a write-up in the Wiki about RAID that gives you a sort of 10-foot perspective on the thing. The RAID tech you invest in will determine how you move forward in the coming years in terms of data protection and storage growth.
And if all you care about is the power cord then you're getting into this for the wrong reasons. IF that's all you care about you could get a DAS solution. You could go with just a simple dual disk dock, or a dumb 'ole JBOD enclosure, or get some kind of real RAID solution, either by buying a card and building an array inside your computer or in some kind of external enclosure, or by buying a device that handles RAID internally and exposes itself to the computer as a single dumb disk.
Currently looking into these two models:
Keep in mind that those are dual-disk models, which limits you to RAID0 and RAID1. This is quite limiting in terms of options, and in terms of volume. However upgrading capacity is cheap, since you only have two disks to replace (however this is only possible in RAID1, and impossible in RAID0 unless you've got enough external storage lying around).
Theoretically, if FreeNAS can read and write to the drive, then the connection type doesn't matter.
In practice, USB adapters like this tend to report the same serial number for every drive, which causes problems because then FreeNAS thinks it's the same drive. Docks like this also aren't intended for 24/7 operation, so things like heat and vibration may be a problem.
The "correct" solution is a proper SAS enclosure, SAS drives, and a SAS HBA, but that's expensive and usually noisy.
A slightly more "correct" but still relatively cheap solution that I've used is a 4-bay eSATA enclosure plus an internal-to-eSATA adapter or a FreeNAS-compatible eSATA HBA (which I don't have a good recommendation at the moment). While the enclosure has a USB mode, this enclosure has that "all drives have the same serial number" problem in USB mode. It works correctly in eSATA mode. Be sure to turn off the "power sync" mode (hold the "sync" button for 2-3 seconds) as it doesn't work reliably and will cause the enclosure to power off. You only have to switch off "power sync" mode once; it remembers the setting.
It all comes down to risk tolerance and budget. If you're okay with increased risk, and budget is the most important thing, and you're okay fiddling with gmultipath as linked above, then go for it.
Please remember that RAID (including RAIDZ/RAIDZ2/mirroring) is not a backup. You still need a backup system (3-2-1). There are a number of ways to lose all your data at once: human error, software corruption bugs like the one fixed in 12.0-U1.1, hardware corruption bugs, flood, fire, theft, spilled drinks, animal urine (pets, or wild animals breaking in), etc.
RAID only provides you availability, if you want to save your data RSYNC backup is better.
I actually have one of this clone (different brand name in my country but same OEM since it looks same) and works pretty well, $139 isn't too expensive, you can even put more disks for other tasks.