It will take some work, but it's perfectly doable.
Enemies with special defenses
The players won't have access to see invisible, flight, magic weapons, or ghost touch. Therefore, if you include enemies that can fly, or have DR/magic, or are intangible, these will be huge challenges, and may be downright impossible. The easy solution is simply to not include such enemies, but the more interesting way is to treat them as nigh-invulnerable enemies that the heroes will have to figure out how to defeat. In a normal Pathfinder game, a ghost is a normal critter with a +2 CR template pasted on top, and requires some minimal preparation to defeat. In a low-magic Pathfinder game, a ghost is a mystery: why did they become a ghost? How can you persuade them to pass on, or at least let the party pass by peacefully? Can you persuade the local priest to perform an exorcism, and will it even work? Instead of "find monster, insert fireball," these types of encounters are now role-playing challenges, because they can't be solved any other way!
Alternatively, you can simply strip out the special defenses from enemies. Pathfinder assumes you have level-appropriate counters to special abilities anyway, so by removing those special defenses, you aren't going too far from the original intent. Adjusting the CR is left as an exercise for the GM, because it's going to take a fair amount of trial-and-error to determine what the right balance is.
Fixing armor class, and other numbers issues
Pathfinder assumes that both attack bonuses and defenses will be augmented by magic items. This partially balances out if the players don't have magic items, but consider giving everyone a +1 bonus to AC and all defenses every four levels. Don't make them pay a feat for it, just give it to them.
While you're at it, give your players bonus XP for the monsters they defeat, by calculating the XP as if the monsters were a higher CR. Since they're operating without magic, every encounter is going to be harder than what the DMG "expects" when it calculates XP per CR.
What will they do with their money?
Your players won't be able to buy gear that personally enhances their ability to make things dead faster, or grant them new solutions. If you keep to the normal loot rules, then the party will have far, far more money than they know what to do with. You have two options here: give them less money, or give them something to do with that money.
Let them invest in mercenary companies or land holdings. Let them become influential in the church, or their hometown, or even their country as their economic might and donations in the right places give them power that they would never be able to take with a sword. Favors in high places give characters some very powerful options.
Recovery after combat
This will require explicit house rules; you'll need to accelerate natural healing (heal a percentage of HP per day instead of a flat amount?), allow Heal checks to do much more than they normally do, grant the local clergy some extremely localized healing powers (they can heal people brought to their church, but not outside of their place of worship), and/or make this a political game rather than a hack-and-slash game.
If everyone's having fun, then it's a good game. It doesn't matter if the characters aren't optimized: as long as they feel like they're making a difference in the world and they're enjoying the game, then you're doing it right. The characters will be balanced, more or less: they all don't have access to magic, so intra-party balance isn't as much of a problem. You'll see that the players lack all of the magic-based solutions that you'd expect in a normal Pathfinder game, and you'll select the enemies more carefully, but things will work out fine. Let your players know that things will be a bit different than normal, and your players will go along with it; they requested this kind of game, after all.
You're the DM, that means that you can change the scenario however you want, and specify how the magic items work however you want.
There are a lot of possible solutions, some of which you've mentioned. Personally I'd probably just reskin the evil-looking crown. The magic part of it could be a circlet, but the last owner added spiky bloody bits because they were Evil with a capital E.
For the dagger, you could have some sort of good order of priests or something of that nature offer the PCs a magic item of similar value in exchange, so they can destroy the evil item. Alternatively, you could simply Rule 0 it to change the requirements for using the dagger to being used in defense of the innocent, or something in that vein. Further possibilities include adding some other equivalent magic item as loot, and making the dagger disappear or lose its magic as soon as its wielder is killed.
In terms of players 'fixing' or 'cleaning' loot they've already found, you can create holy shrines that can remove the darkness from items (a nice example would be a spring or other water source, as immersion of items could be used as a cure).
Best Answer
Arguably, this isn't your problem. In most circumstances, you don't give loot to specific characters - you give it to the party, and let them sort out how it gets distributed.
Yes, this could theoretically lead to one player hogging all the magic items, but that's a solved problem; As long as your players are aware that they can negotiate how stuff gets shared around, they'll generally spot unfair situations, renegotiate as necessary, and generally agree on equitable solutions on their own.
If one player's lack of magic items is dragging the party down, that's the whole party's problem, and most players will realise that pretty quickly; Even the most selfish of players will agree it's a good idea to share the wealth if it improves their own survivability.
If magic items have already made one player much more powerful than the others - Well, it's not like magic items are bound to the first person who picks them up; Your players can always just redistribute them where they'll be more useful.
If your players don't find an uneven distribution of loot to be inequitable - Well, then there's no problem to solve; Everyone's already having fun, and that's what the game is all about.
So, to repeat myself, this isn't your problem: Working out how to distribute loot is just another part of the game. Heck, some players find it one of the most fun parts of the game: Working out how to effectively use available resources is core gameplay, and deciding how magic items should be distributed for best effect is part of that.