A few notes.
You can have it snow all year, without snowing every day. The problem with snowing every day is the snow will just pile up, and gravity will compress it into an ice sheet. No matter how tall your trees are, eventually they will be buried. That is what has happened in Antartica; the ice sheet has an average thickness of two kilometers. Your trees are not two kilometers tall.
Also if it snows every day, photosynthesis is impossible. I'd make the snow intermittent, and I'd add some strong winds to "clean" the leaves (pine needles) so they can photsynthesize.
As for seeds: Not all plants grow from seeds. In particular, see here:
Plants that reproduce from asexual vegetative reproduction
New plants are sometimes made by asexual vegetative reproduction.
These new plants have exactly the same genes as the parent. Some
plants – like strawberries – have stems called stolons that grow out
sideways above the soil, and new plants grow up along them. Other
plants send out underground stems called rhizomes, which form new
plants at a distance from the parent. Tubers (for example, potatoes)
and bulbs (for example, onions) are also special underground
structures that can grow into new plants.
This works for your trees, their roots are under the permafrost, so the rhizomes (underground stems) can spawn a new tree, and feed it, it eventually breaks from the parent and grows to break the permafrost. Just like plants that can crack concrete. Once it breaks through and starts photosynthesizing, it breaks from the rhizome of its parent, and grows independently.
This permafrost is probably cracked anyway (a path to grow through) by the adult trees growing; there is only so much surface area, so as the cross section of a tree slowly increases, the permafrost will be squeezed, push, and buckle. Maybe a good opportunity for the underground baby to push up and break out.
edit: P.S. You might want some sort of underground warmth that can cause the falling snow to melt. That way it wouldn't pile up as ice. Yellowstone Park comes to mind, an ancient volcanic crater that makes the ground warmer than the overlying air. So no permafrost at all. You could still have ground "seasons"; so a "summer" in which the atmosphere can still be -10c, but the ground is +10c so the snow melts and drains into the soil. If you want flowers, then this is the time to bloom: insects that live underground all year (similar to locusts) come out in vast swarms in the ground summer, brave the cold to collect the nectar (pollinate the trees), grow fat, mate and dig back in to lay their eggs for next year, and then starve to death keeping their eggs warm through the coldest part of the winter, leaving their body as food to nourish their hatchlings. (If you think that's weird, the mayfly has a 24 hour lifespan.)
The pollinated flowers is what triggers your trees Rhizomes, growing underground, to actually produce new trees that are not just clones of itself.
Voilá! Flowers, and sexual reproduction! (It's scifi, you don't have to get any more detailed than this, at least it isn't 'magic'.)