If I Connect To Server from a remote location, not on my home network, will that icon appear and will I be able to activate Share Screen from a remote location?
No. But you can still use Finder -> Go -> Connect to server... vnc://your.dynamic.ip.name (VNC) and afp://your.dynamic.ip.name (file sharing)
VNC needs port 5900 port forwarded
AFP needs port 548 port forwarded
I would mention Back-to-My-Mac (BTMM) but I'm not sure if it is still available to Snow Leopard users with all the iCloud changes. If it is, BTMM can deal with all of this without a dynamic DNS name and without opening ports on your router.
Is there another way?
Another options is TeamViewer.com which will allow remote screen sharing without needing to open ports and without needing to get a dynamic DNS name. TeamViewer.com can also be a backup if you want to make sure you have the ability to connect.
I actually have several computers at home behind a router. On the router I had to forward vnc requests to port 5900 on the home computer I want to control.
If you need to connect to other computers, most routers will allow you to port forward a different internet side to 5900 on the home side. Then you would use something like vnc://your.dynamic.ip.name:5901, vnc://your.dynamic.ip.name:5902, vnc://your.dynamic.ip.name:5902, etc... The same trick can be used for 548 file sharing port. The key is that the internet side has the non-standard port, but the router assigns the standard port number on you home LAN side going to the desired computer.
You can also just connect to the 1 computer and then use that computer to connect to the other computers if you absolutely have to. Awkward, but possible to do.
On my home network I was testing with afp://homeipaddress and this connected the two computers so that I could share files (Apple File Protocol).
If by homeipaddress, you mean your.dynamic.ip.name then in theory you were going out to the internet and back in through your router. A better test is to take a laptop to a WiFi hot spot at a cafe, bookstore, etc... and give it a test.
Another way to be sure you are going through the router, is to use the router trick of assigning a different internet side port number and have the router change the port number to 5900 or 548. For example have the router use internet side port 12345 reassigned to port 5900 on your home Mac. Then use another Mac to connect to vnc://your.dynamic.ip.name:12345 and if you connect, you are most definitely going out and back in through the router.
And if you are comfortable using a non-standard port number on the internet side, you will make it more difficult for malware on the internet scanning for open ports to try and break in, as they like to target known ports, so if you use something with 5 digits and under 65535, it is less likely for them to quickly discover your open ports.
And with open ports, make sure you have good passwords on the Mac(s) where those open ports go.
