Starting with the easiest to explain and working to the harder to explain
3. Medieval-ish Social Structure
I want something similar to a feudal society. With analogues to kings, Nobles, knights, and peasants.
Allow me to introduce you to your feudal lord ... er ... the CEO. Considering the power of corporations in America it would not be too big of a stretch to see that after the apocalypse a feudal structure based not on noble titles, but ranks in a corporation.
Interesting enough as corporations are basically people now, it would not be too far-fetched to imagine that the corporation itself becomes something more than its name and guiding principle once the world goes tits up.
Admittedly the corporations of today would have to diversify into doing a bit of everything. In a theoretical future before the end, maybe they did. Corporate towns of old came back into being with a government that allowed it, basically making people serfs in better clothes. After the end, with nothing to stop them, the corporate leaders basically became kings of their fiefdoms.
But really, you'll have corporate CEO-kings that demand everything of their worker-serfs and strive to take as much from them as possible without killing them outright. Bonus dystopia points if being fired in this world is literal.
Side Note: It might be interesting how much, or more accurately how twisted, the US Constitution will end up being in a world like this.
4. Medieval-ish Technology
Basically no technology that couldn’t exist in the 19th century.
Large-scale access to technology needs to go anyways -- we can't have the newfound worker-serfs have anything that could possibly put them even one step closer to the elites. Conveniently, with how much of our technology is based on other technology, it may take a long time to build everything from scratch depending on the level of destruction, decay, and radiation.
Short of every library catching on fire, the theoretical knowledge of technology will remain and hopefully be passed down. In relation to the society, it may be that only a certain class of people get the education that could rebuild technologies and because of that, there is a marked difference in technology based on class and/or it just takes longer to rebuild.
Remember the elites will want to keep their power and will only have an interest in technology insofar as it increases their profits without endangering their situation.
In addition, scavenged relics combined with post-apocalyptic jury-rigging could be the aesthetic. Retained technologies will vary based on available resources and usefulness in daily life.
And since it is America -- yes there will be firearms. Even if they have to hand-forge muskets, they will have guns. The big question is if they could effectively forge bullets. If the skills survive, you may have 19th century firearms.
5. Adventurers
We need people to go on adventures, types of adventures, and reasons to do so.
It's not inconceivable to think that trade wouldn't happen between corporation-states in this after-world. While the caravan might not be adventurers, the guards of the caravans might well be, adventuring to find a place to call home or just because it pays better.
In addition, given that the apocalyptic trigger is nuclear war, there may be areas that fared better than others because they didn't get bombed and the winds were on their side.
A story of a group of adventurers, knowing there might be resources elsewhere that can't be gotten within their lands, venture out into the wilds of the end of the trail known as I91 on the old maps to do trade with the northern mountain people with their strange language and customs.
Today, we call that driving through Vermont to Quebec but after the end, it may be an undertaking.
1. Magic
Fundamentally, your second point of Fantasy Creatures is based on this one, so I have to deal with this first.
Ultimately there are two broad ways to handle this -- magic as lost technology, or magic is trying to exist. As others have touched on the first, I'll hit the second.
Say that the damage caused by the bombs themselves create breaks in the ley lines of the Earth that lied dormant for a long time, allowing that power to seep into the world. But it is not unlimited power, and it too may be tainted by the radiation of the bombs.
With the nuclear war came magic, but we humans don't believe in magic. Even as society regressed, we still did not believe in magic except for one thing -- the invisible force that kills you if you venture into specific areas. Usually the largest ruins and the lands around them.
Because of that one crack in our rational worldview, magic can work on that and that alone. This is why the limited magic in the world works on the EM spectrum and is fairly rigid with rules and the like.
One might think that with enough religious belief, there would be divine miracles but there are not. But even if you have faith in your deity, it is common knowledge that your miracles do not come from ordinary humans. As one example: Christians are told of the miracles of Jesus, but they are also told that Jesus is a demigod of an immensely powerful deity. So no, they will not be miracles from the Churches ... probably.
2. Fantasy Creatures
The world's newfound magic does interact with the radiation from the war and strives to keep as much alive as possible. It's not pretty, but ultimately it is effective.
For us humans, well who knows what's been cooking with our sciences between now and the war? Dwarves could have been one Megacorp's idea before the war to use "precise genetic recombinations to achieve enhanced underground suitability and extended employable time frames". After the war, they live on as what we would consider dwarves.
The smaller peoples started as the suburban kids that were forever changed by the bombs that went off, orphaning many as their parents died in the nuclear fireballs that engulfed the cities. Stunted by both radiation and magic, they either wander or end up exploited by the corporate lords for their small size and physically weaker stature.
In the end, your basic formula is Animal + Mutations + Magic = Fantasy creature. There is even room for much subverting of ideas such as the unicorn's horn only have healing properties for the unicorn because it contains the radiation from the animal's food. I do not want to know what the marine life near and north of Boston would mutate into.
Extra Notes
When planning your nuclear war, do make note of what cities get bombed and what ones may avoid a nuclear fate. While they might be hit by a potential nuclear winter, they might avoid a lot of the actual fallout.
It could end up that smaller cities will be the heart of corporate feudal power.