You ask a lot of question. First and foremost I want you to understand just how effective armor really was: This channel does a lot of testing and here you can find a series called truth about linen gambeson/mail/plate armor. So the point I'm coming towards is: armor is very effective and probably wins more than weaponry from increase in steel quality and therefore will be produced more and/or see more development.
Another thing to consider is that a suit of high quality plate is very expensive and assuming that your improving processes are add-on would more likely make it even more expensive. This by the way brings us to the second most important argument we have to settle:
How exactly your magic works?
Is it a post-process enchanting or a supplement for actual metallurgy processes? Is there a way to use it to not increase the quality but rather optimize the production and opt for more quantity? Or does your furnace dragon only agrees to lend his fire for master-piece craftsmanship?
One of the defining features of the Industrial Revolution is that many things became mass produced and therefore widespread - it is the revolution part, the evolution is increase in quality.
The answer to this question will be the answer you're looking for:
If it's only quality - armor evolves further but only for the rich and knightly classes. You may get master-piece level weapons but again only for the rich and powerful.
If quantity(through better metallurgy) can be achieved - I'd say you'll see a bigger variety of armor and weapon systems because whoever fields more better equiped troops wins.
The second one is more of a game-changer since now you'd be able to field heavy infantry and heavy cavalry that are not knightly class but posses somewhat compareable equipment. A pikebox dressed in brigandine/cuirasses that can repel a charge of knights and advance onto the enemy despite the arrow fire will be an effective force.
Now some direct answers from me with a little back up:
1. Would weapons be lighter?
Mostly not, because weight is not a drawback for most weapons.
1b If so, would typical longswords be replaced with sabers or rapiers?
Longsword did not evolve into repiers or sabres - Rapiers and Sabres became widespread after the role of armor diminished and are optimized against un- or lightly armored oponnents: one is for slashing, the other is for thrusting attacks.
However better steel may result in that swords with sharper edges or different crossections may become more viable.
2. Would axes have greater penetrative power?/Would projectile weapons become more penetrative?
Only a marginal one, not enough to make a diffirence. Even a two-handed pole-arm with a spike doesn't guarantee that you'll penetrate a plate armor, hence why it dissapeared from a polaxe and the hammerhead on it became wider and with a bigger number of mini-heads to not glance off and better transfer the energy rather than attempt to penetrate or dent it.
If we talk about projectiles: you have to understand that Arqebus and Muskets advanced to a whole new level compared to Bows and Crossbows in terms of armor penetration capabilities: a 60g bolt that flies at 40m/s produces 48 Joules of energy, a 35g bullet that flies at 180m/s produces 567 Joules of energy(more than 10 times!).
3. Would there be armors with higher mobility?
If only slightly. The effect of weight through better quality is miniscule unless you want to dab into the territory of specialized alloys.
The mobility part depends more on the fine craftsmanship rather then materials. The "precision boring, honing, and other machining tools" which you ruled out would've helped much more in this area.
4. Could bows and crossbows be given higher draw weights?
Draw weight is not the key factor, it is one of many but was reffered more to because of the lack of scientific understanding behind the damaging effects of the projectile. Here's an example of a modern 180LB composite crossbow and 70LB compound bow outperforming a medieval esque 300LB crossbow the answer for this is Kinetic Energy which cares about projectile mass and velocity(more so velocity hence why firearms are so much more effective with their 200-900m/s).
Draw weight is simply one of the ways to achieve more speed. Basically it's like "fuel consumption" if we make an analogy. Modern composite matierials or technologies(compund disc) allow to better translate the power of the draw into the speed than the steel.
But given your assumption better quality may result in bows and crossbows becoming more deadly(better velocities, longer range etc) if the metallurgy improves the right qualities such as yield strength. However don't expect the results be as groundbreaking as even early arquebus.
5. Would certain types of armor be made irrelevant?
If you can optimize the production processes Mail armor will become obsolete because Plate provides better protection and is now easier to manufacture(Mail requires a lot of manual labor and time to produce). The whole concept of interlocking rings developed because metallurgy could not produce solid plates of required size for a long time. It will still remain as protection for joints though: neck, armpits, groin etc because of the flexibility it provides.
Brigandine and other Coat of Plates types of armor may become either obsolete or insanely widespread depending on how optimized the production is. If solid plate of big size is easy to produce your world will go straight into full body breastplates and cuirasses, if it is not an easy process Coat of Plates will be the way to go and quickly replace Mail armor for those who can't afford full plate.
6. If armor were stronger would it make certain weapons ineffective
Thing is armor already makes some of the weapons obsolete, but there are ways to work around given enough skill and luck. The question is how widespread your armor is to actually rule out weapon classes. However I doubt that levied peasants will be able to afford a suit of full plate to rule out edged weapons as at least sidearms.
7. Would the armor protect from death more often and lead to more focus on disabling and wounding enemies than on killing them
This is exactly what happened. However on a grand scale of things I'd rather call it "diversification" rather than "supplementation" that you hinted with the "than" in your question.
8. Would small groups of skilled warriors become more effective with better equipment?
Small groups always benefit the most from more skill and better equipment. Hence why we have all the spec-ops buzz. But again you have to understand that better steel won't make them into super-humans that can kill 40 people in one battle and then do it again the next day. A Machine Gun though...
9. Would medical treatments change? Could better steel improve medical treatment? (some magic is used for minor healing). Can't say for the surgeon part of the question but the last sentence may be a game changer depending on how you define magical treatment. If magic killing bacterias and viruses in the wound is considered to be minor(nothing like insta heals we have in RPG's right?) - I can envision more open but easier to mass produce cuirasses, sallets, burgonets and morions becoming widespread, because it will be possible to outfit heavy infatry/cavalry with armor protecting only the essential bodyparts for much cheaper and still guarantee surviveability because damage to the extremeties won't be as fatal as they used to be.
So to summ it all up: if you want more effect on the world you have to either opt for much greater quality through alloys(second half of 20th century and beyond) or more optimized production(like with dragons replacing Blast Furnaces and reaching better temperature at that) that will make the end products more widespread and affordable.