Ultimately I think it is more optimized to take a background that gives you a skill and skill feat that is unrelated to medicine, medic, and Chirurgeon. It won't really cost you anything, but it will open up new sideline options for your character.
Basically, what did your character do before they became an alchemist?
And yes, the Crafting replacement of Medicine from Chirurgeon is not fully handled in the rules. I expect that it will get officially tweaked in errata at some point. For now, tweaking it yourself in your games would probably be fine.
TLDR:
Is their a rule abiding way for a chirurgeon the spent the points in skill points in medicine from the background and Medic Dedication somewhere else?
Unfortunately, there's none. For the background, it's still quite easy to choose one that doesn't give Medicine, even if it can delay your access to Medicine feats like Battle Medicine.
For Medic Archetype, on the other hand, there's absolutely nothing you can do. If you want to get it's Medicine related benefits, you'll lose the bonus proficiency to Medicine. On the other hand, it's not the only case where a Dedication doesn't give you all its benefit due to your class. So it's annoying but not that impactful. And Medic Archetype fits the Chirurgeon so well that you will not really feel the loss.
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As a tag along on a similar line of thought: sometimes there's an ability that, for example, gives you Nature for Natural Medicine. How does any skills or effect (skill feats I guess I mean) like: Ward Medic, Continual Recovery, etc interact with this? Especially because some are like "Expert in Medicine" etc, how could I get these using my nature?
Natural medicine is a trap feat, since you still need to increase medicine to get the skill feats and target higher DCs
Well, not a trap for the right build: if you have a character that's going to raise Nature anyway [say a primal caster], it can make sense to take it. You only have to buy a single item for bonuses [say a Druid's Vestment] and it comes with a free +2 bonus in wilderness. You just have to know the limitations going in.
Natural medicine feat is not to substitute medicine is just a way to give Treat Wounds for builds that don't really want to focus in medicine but find useful to have Treat Wounds. But I agree that once medicine is a Wis based skill and usually those who takes Nature are Wis based classes so it's probably interesting to they to take medicine too what usually inviabilize that Natural medicine investment.
But if you are in a case that you already have Nature due a background or class skill like the Rangers may be interesting to take Natural medicine to have access to Treat Wounds without need to invest in medicine skill. You won't be a full efficient medic but still can be too useful many times.
Natural medicine is a trap feat, since you still need to increase medicine to get the skill feats and target higher DCs
It's been errataed, you can now target higher DCs.
As for skill feats, they only need the Expert Proficiency to Medicine. So if you want to be an efficient out of combat healer without having to pay for Legendary Medicine, you can just increase it to Expert (with Human Skilled Heritage for example) and focus on Nature instead.In my opinion, it's nice, it allows a party to have an ideal out of combat healer without getting higher than Expert Medicine.