A ship can apparently fire out through it's own shields.
And transport, too - in "A Taste of Armageddon", the very episode to establish that shields and transporters don't mix, Scotty first says he won't be dropping his shields, and the villain then swears he will fire at the ship the moment the shields drop - and Ambassador Fox then beams down, and the ship is
not fired upon.
Curiously, no TOS episode ever contradicts the idea that beaming out through shields would be impossible or even difficult. It's the beaming in that is impossible, meaning anybody who goes out will be unable to get back in, which is why our heroes worry about shields vs. transporters.
TNG tries to muddle the issue in "Datalore", where Lore's silly plan of beaming out a tree involves the dropping of shields. But we know it's not Lore's real plan, and we know Picard and friends don't buy it anyway, and we again get a villain interested in doing damage the moment the shields drop yet no damage is done. Beyond that, the TNG era shows, too, are in agreement with the idea that getting out through shields is trivial in all situations, no matter what the means. VOY just inserts a case or two of a hero being beamed in while combat is ongoing...
My understanding of shields was that the frequencies stated were just a device to allow weapons fire to pass out from the ship while blocking incoming enemy fire, unless you happened to match the frequency (the Duras sisters in Generations for example). A phaser blast or photon launch takes only a fraction of the time that a transport beam does, which would perhaps expose a greater window of opportunity to allow an enemy to discover this frequency.
One would then assume the frequency is changed after each outgoing shot, yet ST:GEN sort of suggests this does not happen, by giving the frequency as a single figure. Although it might actually suggest the exact opposite: when shields prove ineffective, the heroes should forcibly change their frequency in addition to all the automatic changing going in - but LaForge immediately relaying the new frequency to the villains would hide and negate all changing.
...in dialogue it seems to indicate that lowering the shields would leave the ship defenceless while beaming (Arena).
"Arena" is telling in the sense that the Gorn ship is supposedly far away, and couldn't easily flank Sulu's ship. Then again, Sulu has just been informed there's a Gorn ambush down on the surface - so there are good odds that the beams coming from the direction of the planet would include not just our heroes' transporter beam but Gorn death rays as well!
As for the displays, the TWoK/TSfS one does consist of "segments" or dots, random ones of which go dark when the shields are compromised - but that's more like the display fritzing out than the actual shields behaving in a certain manner. In TUC, though, Chang is consistently striking under the belt, or then hitting the top of the saucer - so a top-down view might well show "uniform" weakening even though the side shields are much stronger than the top and bottom ones.
Timo Saloniemi