Here's the answers your questions in order of appearance.
- "would a standard router be OK? Are they basically the same?"
The short answer is yes on both counts. The model number you provided is a router that has been branded with "cable" to make it easier for end users to identify what they need if they have cable internet. This was a common thing to do back in 2000 when this product was initially released. Nowadays, items that combine modem and router functions should be called "residential gateways", but those are normally branded by your service provider (like Verizon FiOS).
- "will this effectively replace the BEFSR41 "cable" modem and give me gigabit speed from the machines that support it?"
Again, the short answer is yes on bot counts. The D-link device does perform all the routing functions you need as well as adding you gigabit communications.
- "Can I do better than that with that new router? "
This is a little more ambiguous and I can only say "possibly". The problem with file transfer performance is that there are multiple factors that determine speed. The communication speed between nodes (computers) is only part of the performance equation.
- "Is the performance of the NAS200 going to dictate the performance of the rest of the LAN here?"
The only time the NAS speed comes into play is when you are sending or receiving files from the NAS. Since the NAS can only communicate at 10/100 speeds, there is a limit to how much data it can transmit at any time. Communication between your other nodes can still take advantage of the higher speed communication assuming they are gigabit enabled.
- "Do I need to consider anything else?"
To make it work? No. To have a measurable speed increase? Yes. Performance is determined by a collection of items working together. The communication protocol (gigabit), network hardware (router), and end points (computer/NAS) all can drastically affect performance. The speed of the entire chain is only as fast as the slowest link.
So in review:
Yes, you could replace the router, but it's tough to say how much of a difference it will make. If you plan on upgrading everything to gigabit, then this would be a good device to use if you don't mind the roughly $100 price tag. If price is an issue, you can always pick up a switch like this one. That would still give you the option of gigabit if you plug all your devices into it and connect the 5th port to the router. That would add another device to your network which you might not have room for.
In my personal opinion, I would buy the router you have been looking at and replace what you have. The linksys device is 10-ish years old now (though the device itself might be younger), and was designed to meet the needs of consumers then. Between then and now, we've had a number of improvements to networking technology that could make your life easier. From what the reviewers say, this is a feature rich device that will do all sorts of things that you will probably not use. That being said, it will allow your network to be prepared for future purchases. New devices will most likely have gigabit network cards installed in them which will take advantage of the increased transfer speeds.
If you add the device and you don't see a change in performance, then something else has a bottle neck and will need to be upgraded to increase speed. That is a bit outside of the scope for your questions.
Hope this helps.
Don't forget that you might need new network cables to get gigabit speeds. Cat5e is what you are looking for. you have to be careful that you don't get ripped off by a big box store. I wouldn't pay more then $5 for a 25ft Cat5e cable. You don't need Cat6. More info here.