I think that some Dalish might be the 'Take no prisoners!' type, but not all. I also think that it would start out with guerrilla warfare, allowing the Dalish to roflpwn the human forces and disappear, causing chaos in the military ranks as they won't know when they'll be hit, let alone where and how hard. I suspect many soldiers will be losing sleep over it, morale will drop, some soldiers may even desert. Some leaders may resort to burning down entire forests to weed out the Dalish clans, but I suspect that the Dalish would be long gone by the time the fires reached them. The Dalish would probably find sympathizers among the city elves, especially in the Dales, so there could riots and other forms of insurgency that would weaken the human nations considerably. Ferelden might be saved from conflict with the Dalish if Alistair is in charge, since his giving the Dalish land in southern Ferelden would've improved relations between Ferelden and the clans. Orlais would likely be the first to fall, since they were still recovering from the Fade Tears and a civil war.
Rivain might be one of the last nations to come to open conflict with the Dalish due to their relatively friendly relations with the Dalish clans, but since the Qunari get a steady supply of viddithari from Rivain, it is possible that the Dalish conquering Rivain could cause relations between the Dalish and the Qunari to sour, even if they team up to conquer Tevinter. I'd think that the Dalish would wish to avoid provoking the Qunari, knowing their strength, so Rivain might be spared from the Dalish advance.
Nevarra is difficult, since they have powerful necromancers and have clashed with Orlais multiple times in the past, often beating them. Really, I could see Nevarra being one of the last nations to withstand the Dalish onslaught.
The Anderfels is under constant threat from the darkspawn, much like Orzammar and Kal-Sharok. Not to mention that it is the home of the Wardens. Not even the Dalish would want to cross the Order, so it would be likely that the Anderfels would be relatively untouched. And since the typical Anders looks towards the Wardens to rule, I doubt that they'd join in the fight against the Dalish unless directly attacked. The Wardens' policy would likely be to avoid conflict with the Dalish if possible. Some Dalish might be willing to fight the Wardens, but most Dalish have too much respect for them to be willing to anger them, so it would likely be a small minority.
Antiva is also tricky. It is the home of the Crows, so it would likely be spared from the fighting at first, but once the Dalish become more bold, they could possibly send a few guerrilla strikes to probe the Antivan defenses. Since the Crows specialize in assassinations, not all-out fighting, I'd assume that the Crows would be at a disadvantage simply due to not being used to being on the receiving end of surprise attacks. It is also likely that there would be defections, resulting in the Crows to be weakened and the Dalish possibly gaining knowledge on the Crows' tactics and mode of operations.
As for Tevinter, I could see the Dalish pausing to gather up their forces before deciding to invade, again sticking to guerrilla tactics to keep the magisters off-balance and confused. I could potentially see the Qunari deciding to move in from the north to take care of their long time enemy, but I'm not entirely sure if they would. Maybe the Dalish would send delegates to Seheron or Par Vollen in an attempt to negotiation for assistance, allowing the Qunari to potentially keep whatever land they conquered. The Dalish would likely want to count the Qunari as allies rather than enemies, so that sort of arrangement would be generous considering how the Dalish are keeping the land they conquered for themselves. Minrathous woud likely be Tevinter's last stand, since it is heavily defended and fortified, ready for a battle of attrition. The Dalish might also send delegates to Orzammar and Kal-Sharok, promising assistance in clearing the darkspawn from the Deep Roads in exchange for lyrium and an embargo on Minrathous, essentially cutting the city's supply-line. I imagine that once Minrathous is conquered, the city will be split between Qunari control and Dalish control much like Berlin was post-WWII.
All in all, I'd think that Ferelden, Rivain, and the Anderfels would be the sole surviving human nations, Orlais, Nevarra, Tevinter, Antiva, and the Free Marches all being conquered. The surviving nations would likely try to avoid conflict with the Dalish afterwards, seeing that even if they united, they would likely be crushed. It is possible that once the Dalish conquer Val Royueax, the Anderfels would attempt to fight the Dalish, but I don't think that the First Warden would call for action as long as the Dalish didn't actively attack Warden compounds. I think that the Dalish might send delegates to whatever Warden compounds they encounter and negotiate ways for them to avoid conflict.
As for how the Dalish would interact with the Qunari afterwards, I'd think that it'd be similar to the Cold War post-WWII, the Dalish and Qunari both finding ways to one-up each other with their relations being strained at best. The Dalish would likely search for ways to enhance their magical capabilities and the Qunari focusing on their black powder and weaponry. It is possible for Rivain to eventually become a part of the Qun, giving the Qunari a fairly decent portion of additional land. The Dalish would likely also focus on building up their numbers so they won't be spread as thin, so I can imagine them possibly conscripting people into service.
Anyway, I think that if the Dalish managed to conquer most of Thedas, only a portion of the human nations would remain along with the Qunari, leaving both the Dalish and the Qunari as the 2 main powers in Thedas. The Dalish would probably leave the remaining human nations alone for the most part, trading with them and the dwarves occasionally. Besides Tevinter, Nevarra and Antiva would be the 2 most difficult nations for the Dalish to conquer. Relations with the Qunari would be shaky afterwards and both the Dalish and the Qunari would engage in a cold war with each other, eventually ending up in all-out war as the Qunari invade. It would depend on how long the Qunari waited before invading for me to tell the outcome of their invasion. They managed to conquer half of Thedas on their own and they went in blind. If they tried again, it would depend on what countermeasures the Dalish had developed to determine who would win.