The only notable area in which the Dominion seems to have an advantage over the Feds is ship construction which was stated to be faster.
Other than that, the Feds are also perfectly capable of cloning and extensive genetic manipulation.
Problem is ... they don't utilize such measures in wars.
As for the area the Dominion occupies and how many ships was fighting in the AQ...
Keep in mind that there were convoys coming in from the wormhole for some time before the war started and the Dominion gained a foothold in the AQ by allying themselves with the Cardassians who lend their military to the war effort.
In numerous battles the Dominion did not face the Feds on their own.
In 'Sacrifice of Angels' the battle was between Dominion/Cardassian forces versus SF ships only (which were outnumbered 2 to 1) that managed to hold their ground in combat even after suffering some losses ... and then the Klingons intervened which opened up a hole the Defiant utilized to get to DS9.
The capabilities the Dominion showed were nothing too impressive (aside from their faster ship construction) when compared to SF and the Federation.
Their phased polaron beams as I stated were not passing through the Defiants shields in 'The Search' and that ship was carrying anti-Borg technology (something that was supposed to have been introduced 14 months after BOBW and then presumably delayed ... which was stated for the Defiant because the Borg threat was apparently not emminent).
So the fact that SF got their hands on the bug ship doesn't automatically insinuate they were able to circcumvent their weapons like that (although it's a very good possibility).
The tech was probably there before just never utilized (you know that SF was portrayed in this idiotic fashion before).
Given the fact the Dominion never fought on their own in large battles against combined forces of the Federation/Klingons/Romulans but with help ... it stands to reason they would most likely be a tad smaller or the same like the Federation.
It's possible their space is larger as well, but it was never insinuated nor confirmed.
Plus if SF was in Dominions position, even they would have additional 2000 ships ready to come through the wormhole in the mid of the war after gaining an ally and securing their economy to their needs into another quadrant to fight.