OK, looks like you have done some pretty good troubleshooting. Let's go through some steps to see if everything is set up correctly. This is the way I like to do it. It isn't necessarily correct or secure, lol. Once it is working, you can make it more secure if you wish.
It would be very helpful to know the name and model of the modem. It would also be helpful to know what OS you are using for the clients and main computer. Hopefully, no one is using the "main computer" but that is a topic for a different day.
1. The main computer should have a static IP. When it goes to renew its IP address, there is always the chance the DHCP server will change its IP. If you are connecting via the name, this shouldn't necessarily matter, but the first thing to do is make it static.
2. How to do: If you are using Windows 7 Pro (hopefully and not Home), then go to Control Panel | Network and Sharing Center. Select Local Area Connect or you can select Change adapter settings and left click on the NIC. Once open, select Properties | (may as well make sure speed is 1.0 Gbps) before selecting Properties | Select TCP/IPv4 | Change to Use the following IP address. Enter an IP address which is on the same subnet as the other computers such as 192.168.1.2 or 16.2 or 0.2 (whatever the 3rd octet is for your subnet. Since your router will almost always be x.1, I like x.2 for the server (main computer). Subnet 255.255.255.0 and the Default gateway should be the LAN side of your router (probably 192.168.1.1, or again, whatever the subnet is. Set the DNS for the IP address you entered above. Click OK, OK and Close
3. On the AC folder on main computer, set the following permissions to Everyone for Share with full access and for Security, use Everyone and set to full. OK. The share settings is Microsoft's preference. Sure Everyone for the permissions is not the most secure setting, but who cares for now. Keep it simple.
4. Personally, I don't like using a modem as a router/firewall. It is not likely the issue, but if you are going to use the new modem/router/firewall, I would change the settings so it is just the modem, then use a separate decent router/firewall. I also use a switch and now all of the clients are using just the switch to connect. If you do this, make sure the switch is a 1 Gbps switch and if you are going to purchase one, get a Cisco or Linksys or Netgear and spend the extra bucks to get a managed one. If you are thinking of VoIP down the road, get PoE. But, make sure it is managed. Just will make things easier down the road.
5. The main rule in troubleshooting is to reduce the issue to as few problem points, i.e. connect one computer directly to the main computer either using a small switch or crossover cable. A lot of the newer patch cables will allow you to use it directly. If you do this, you have a wired connection that eliminated the modem/router/firewall/switch, etc. Once that is working, you can go backwards to the actual setup.
6. It sounds to me since you can access the main computer and, hopefully, the AC folder, it is probably the .xml file itself. In the main computer, find the .xml file "AmazingCharts.xml" and open in a text file by changing the extension to .txt. Make sure the database path is set to the local path of Amazing Charts, so it may be >C:\Program Files (x86)\Amazing Charts\AmazingCharts.mdf< in bold. Make sure the last part of it is the actual Amazingcharts.mdf database. Just glance at the port to insure it is 61067.
7. When you open AC on the clients and browse to the .xml file, MAKE SURE, you are browsing to the main computer's .xml file and not to the local .xml file. Sometimes they aren't even in there. I remove them if they are. Also be sure, that SQL Server Express 2005 is only on the main computer. Of course, if you were browsing to SQL on a local client by mistake, you would be connecting.
8. When you first open an AC client on a client computer, when it gets to the login screen (if you can get there), right click on the fairly large white strip at the top where your Practice Name is, and it will show you the path it is using. You can change the path from there if you wish. You can also do this with AmazingUtilities.
So change the permissions, be sure the .xml file in AC on main computer is set to the path of AC folder and the AC database .mdf database. Double check the port number which will most likely be correct. Start with one computer only. If you need to, get a small four-port cheap switch and connect directly from one client to the main computer.
Given you can browse to the main server and change files, your network settings are likely correct. That really only leaves the .xml file and permissions. I would be on permissions. When you browse to the main computer, can you change files IN THE AC FOLDER? If not, it is a permissions issue on the AC folder.
Hope this helps.