"All too easy."
These are some observations I have made about the various gate animations, some things change on an episode by episode basis so dont hold me to the following.
As symbols are dialed on the DHD, the dialed symbol appears on and then rotates around the inner ring of the Stargate starting at the last chevron to be locked or if it is the first to be dialed then it appears at the top center symbol slot i.e. under chevron 7. Only one symbol can rotate around at a time, the rest get queued.
In theory this means you can find gate addresses by looking at the symbols on an active stargate since we know the order they go in by the chevron order however in practise this doesnt really work or at least its very hard work as theres only a brief window of opportunity to see the 7 dialed symbols. You see after the last chevron is locked the entire inner ring lights up and the dialed symbols get replaced with the default symbol that goes under each chevron so the address is no longer visible.
In 1x17 when they dial Orins planet at around 6mins you can see all the symbols for his planet. then the 7th chevron locks and the event horizon appears. as the rest of the glyphs glow the symbols under the chevrons change to their original symbols and not those of Orins planet.
You can see the inner ring of the gate fully lit in the background of the SGAC during several episodes although you dont usually see the same with offworld gates. Sometimes the chevrons are lit as well. Maybe this ties in with the gate shutdown bit I mention below i.e. sometimes the gate doesnt fully power down after the wormhole has closed.
In 1x03 near the end when the creature is in the gateroom the Stargate is connected to the Barren Wasteland but only the dialed symbols are lit.
Sometimes you can see the same symbol being used during the rotation in which case you definately can't pull the address off the gate.
In 1x03 when the creature starts dialing a planet at around 16mins (well at this point you dont know its the creature) you can see all the symbols it dials on the dhd. The first shot you see is of the first two symbols. Then it jumps to a shot of the gate. You can clearly see the first one of the two dhd symbols under the first chevron on the stargate, in this case the first of that address is 16.
Outgoing
Incoming
Symbol Locking
Gate Shutdown