I have decided that my campaign is set in a world of permanent winter, imposed by a (presumably, i love liches) dead wizard. Currently, the greatest threat to the world at large is a warlord from a small mountain kingdom that nobody cared about until he launched massive raids with his army of orcs, goblinoids, frost giants, and white dragons. Also, the yetis in the mountain passes are becoming so aggressive that trade routes in the mountains are nigh-unusable. Any ideas for cool adventures. B.t.w. the monk's master was directly killed by the warlord, which may be useful.

I think it's easier to come up with "cool adventure ideas" ( pun totally intended ), if you have a more detailed and rigorous picture of your campaign world. If you can answer detailed questions about your setting, some of the details will likely practically scream "adventure here!".

Some questions that comes to mind reading your description:

• What is the ultimate cause of the world being in freeze? If it's a Lich, why does she want the world frozen ( it's a lot of work, you don't do that sort of thing on a whim, or just because you hate being too hot )? How is it being accomplished? How might it be undone?
• Where are the Players? What land? What is the culture like? How is it ruled? What are its relationships with its neighbors?
• The Mountain Kingdom, what is it like ( culture, power structure, it seems to be at war with its neighbors )? While you're at it, give all your lands names.
• What has changed with the Mountain Kingdom recently, that they've gone full on expansionist aggression? There has to be a reason for it.
• What is unifying the forces of the Mountain Kingdom? Goblinoids, Giants, and Dragons working together? There has to be something unifying these, as they don't normally work together. What on earth is doing that?
• What has go the Yeti's stirred up? Is it related to the other conflict? Is it independent?
• What are the monastic traditions in your world like ( at least for the tradition the Monk Character comes from )? Is there a rigid structure, and now the school is thrown in disarray? Is it more an individual student/master style, and now the Monk must seek revenge?

• What is the Party like? Who are they, what motivates them ( adventure ideas are usually created with the Party in mind )?

The answers are likely to give you all sorts of ideas.

I love the idea of the arctic, there's so much you can play around with blizzards and extreme cold. But I 100% agree with Vedexent, once you start to flesh those aspects out (maybe you already have) then ideas will begin to present themselves. Personally I always begin with a map. It's a super useful way of keeping up with your world and staying consistent. For example I just made the map below in less than 2 mins with Azgaar's Fantasy Map Generator.

Once you've got that you throw it in paint or Photoshop (whatever you have access to) and you split it up and name some key locations. Make a few notes about your locations, cities and landmarks in Google Docs and you're onto something. To try and aid in thinking up some campaign starters, here's a handful that spring to mind:

• The party have been assigned a task to locate and capture an Orc general alive and transport him back to the city of resistance for information extraction, by any means necessary (some fun RP opportunities here). 
• A small village nearby the local mountain has been increasingly attacked by Yetis, they've never been so bold as to come into town before but now they're fearlessly entering at random times, dragging villagers off into the mountains. The party must deal with this while the army is away at war, find the villagers and work out why the yetis are acting this way?
• Reports have been coming in that strange blizzards have began sweeping through the south of the country, although blizzards are not uncommon here, there is something supernatural about these storms. They sweep through at night unexpectedly, and by the time they clear everybody is the town has vanished. When outsiders venture in days later, all they find is a ghost town.

Oh man. This is a combination of D&D with post-apocalypse. Think about it--sudden permanent winter is going to just destroy so many cultures and civilizations. Here's what I'm thinking:

• Entire frozen-over cities, with the remaining population left fighting for survival, hacking out places to live from ice-covered buildings, trying to find anything that will burn to keep warm. 
• Desert nomads driven out of their homeland, turning to pacts with dark powers to keep their people alive as they are forced to invade nearby nations for survival.
• Aquatic elves trapped under the ice in what used to be open ocean.
• Crop failures and famine are rampant. Isolated pockets of people are turning to cannibalism.
• Draft animals are dying off due to the cold--refugees are often enslaved to work fields.
• A hidden city inside a glacier, barbarians already from the north lands but driven into hiding, and slowly learning the ways of ice magic.

You’ll definitely want to research life in Siberia during the late 1800s and early 1900s. That will give you an idea of just how inhospitable an ice-world of frozen tundra would be. It’s a cool idea. I’ve been eyeing the Spine of the World in Northern Sword Coast for some adventures for my table. Even better is that it is now just Nightal day 11, in essence December 11th, so winter is just taking hold! Brrr!

Quote from Cardinal_Fang >>

You’ll definitely want to research life in Siberia during the late 1800s and early 1900s.  

Building on this, you should also check out the show Fortitude for inspiration. It's set in Svalbard, one of the coldest places to live on the planet! It's got an incredible plot too!

Great ideas! More would be welcome, but i’m going for a classic pulp fiction feel, so I don’t need much character development beyond: “he’s insane and evil, so screw motivations”

I think that the desperation of a frozen world would lead many to justify insane actions and evil intents. Cannibalism was a thing in isolated areas b/c of such desperation. 

Also, research the Donner Party if you have a few minutes. RL accounts of humanity being taken to and beyond the brink of crazy, irrational behavior. 

In particular regarding cannibalism, I'm thinking of two neighboring cities. Used to have great relationships, but since the eternal winter, things have gone bad. They began warring with each other for limited resources, and the leaders realized that they'd all just kill each other eventually. So they made a bargain. Each city now by itself seems fine, and there's peace between the cities. But that's because each year (or month, depending on the city size and how you want to parcel it out), each city sends the other one lottery-selected citizens for food. :)

As resources became scarce, neither city could face the prospect of eating their own people. And they were all tired of the war. So they made a deal. Now each city eats the other's citizens. And there's peace.

Food could be a good motivation fr the warlord, or maybe he is the scion of orcus