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Personally I choose to believe that Aroden was becoming an evil, human supremacist deity, so a cabal of other gods made a move against him. Think about it, slavery (especially of the other, "lesser" races) was common in Azlant. These beliefs made it to Old Cheliax, surely due to the "God of Humanity" as he was called. Hell, the Starfall Doctrine even talks about that(Old) Cheliax would become the pre-eminent nation of the world and he would lead humans to a newfound age of peace and justice. I think it'd be highly likely that the precursor of that would've been a lot of slavery and one of two genocides.

While Calculated Splash is very good, it's not required. By level 20, assuming max Int, it's a 3 point damage increase. When you can get it, it's a 2 point increase on your highest level bombs, and stays that way up until level 20, with it being a 3 point increase on the Perpetual ones.

Is it good for a bomber alchemist? Definitely, in the same way a one-handed fighter wants Dueling Parry. Is it required? I'd say no.

Yeah, too bad that at level 10 you're gonna want to take Expanded Splash to make the base splash damage stack with your int. So it actually takes 2 feats to get int to your bomb damage and your int mod should be high enough to make it a no-brainer if you want that or not. In addition Expanded Splash increases your Splash Radius. So any Bomber worth their salt will want Expanded Splash, with Calculated Splash being a feat tax for it.

I think the following on Subordinate Action on p462 of the CRB is relevant:

Subordinate Actions
An action might allow you to use a simpler action—usually one of the Basic Actions on page 469—in a different circumstance or with different effects. This subordinate action still has its normal traits and effects, but is modified in any ways listed in the larger action. For example, an activity that tells you to Stride up to half your Speed alters the normal distance you can move in a Stride. The Stride would still have the move trait, would still trigger reactions that occur based on movement, and so on. The subordinate action doesn’t gain any of the traits of the larger action unless specified. The action that allows you to use a subordinate action doesn’t require you to spend more actions or reactions to do so; that cost is already factored in.

So Elf Step makes the Step be 5 feet in any case, as it is a specific Modifikation.

Alchemist feats before level 10-12, excluding bomber, are sad really.

Even those Bomber feats are largely of the Kind Paizo wanted to move away from with 2e, that being math fixes. Every Bomber is expected to Grab Quick Bomb so you dont have terrible Action economy when bombing. Hell, your bombs damage does Not Even Scale by default with int. Instead you get 2 feats that first add int Instead of splash damage and the you get both. There is no choice involved, you better take both so you can keep up with higher level Fights.

What is a (minor) issue however is ... 'Access You are from Old Cheliax'

Why? Because there's Citadel Vraid sitting nearby Korvosa in Varisia. It houses members of the Order of the Nail. I'd have thought it would be ok to become a member of that order without having been born in Old Cheliax. Surely they take the occasional recruit from Korvosa, someone who has Chelish ancestors but was born and bred in Varisia?

Perhaps it's all in the interpretation of the word 'from'?

Old Cheliax is one of the Meta Regions they mention in the CRB. Per the map they showed earlier in the year Korvosa is Part of that Region, even if it is on outer edges of said Meta Region.

You could also grab Far Shot to double your range increments. It's another feat to make alchemical xbow usable though.

You only ever get 1 extra Action from quickened. If there are multiple source for the effect you’re able to choose from any of the allowed Action when you use your quickened Action. So if you have a weapon Rune that allows you to Strike and a spell that allows you to stride with your quickened Action, you’re allowed to choose between striding and striding when you use your quickened Action. You never gain 2 quickened Action because the quickened does Not Stack with itself apart from allowing other kinds of Action.

I want my Familiar and Druid Dedication will give me cantrips and solve as quickly as level 2 the "peasant with a crossbow issue" ;)

I'm not sure if attack cantrips are a solution to the "peasant with a crossbow issue" unless you go with an int caster. Druid keys of Wisdom, so both their Spell Attack Rolls (spellcasting ability modifier + proficiency bonus + other bonuses + penalties) as well as their cantrip damage are dependent on their Wisdom score. So it's yet another Stat you may need to increase, plus the Archetype feats for Expert/Master Spellcasting to keep up your proficiency. So not exactly cheap to do.

And bombs have other drawbacks:
- You need high Dexterity.

I'm unsure as to how this is a drawback. Dex also provides armor for you, so it's quite good considering Alchemists only get light armor. Going melee means you either take dex anyways (finesse weapon) or you go strength, where you at least want moderate dex so your armor doesn't suck. Or are you talking about multiclassing into a spellcasting class?

What really grates me is that the perpetual infusion situation for Mutagenist and Chirurgeon isn't even salvageable with errata due to it being ingrained in the class progression. If instead of the perpetual infusion being ingrained into the classes chassis we'd instead get 5 Discoveries we could just replace those not very useful perpetual infusions with something else for Mutagenist and Chirurgeon. Instead we now have the situation where each single Alchemist is forced to have perpetual infusions, unless they take an archetype that replaces them. In my opinion they designed themselves into a corner with this class feature.

I have to be honest, as the alchemist currently stands I can't see myself ever going full alchemist. The class is just not built to rely on its own resources and abilities on the long run, so you'd better mix it up.
Wanna be a bomber? Go Ranger Dedication, you could even get Improved Far Lobber (Far Shot at level 8) and Improved Quick Bomber (Quick Draw at level 4) out of it. Of course the main point would be to rely on those nice Crossbow feats the Ranger has once your resources run dry. Hell, despite it normally costing you one interact action to grab a bomb you could still use stuff like Hunter's Aim with them if you really need that +2 to attack. Plus you can get running reload, which means you even get a free movement action when you reload that crossbow, which means movement to your allies to apply elixirs.

As an Chirurgeon I'd probably go with a very similar approach - grab those high mobility ranger options (Running Reload at 8, Skirmish Strike at 12) and apply those Elixirs. Not sure what else I'd do as a Chirurgeon, its perpetuals are sadly far too situational to really get use out of. I'm really disappointed in this.

As a Mutagenist I'd first have to come up with a build that overcomes my MADness, so I'll probably drop Int to at least 16, if not even 14. They really should have added Strength as a key ability score option here, currently it really seems like a trap. Having Finesse on the Bestial Mutagen attacks would've been great for less MADness aswell. But realistically I don't even want to use the Bestial Mutagen anyways, the defensive drawbacks seem too big too me (even moreso once the AC gets reduced to -2). Instead I'd grab a single one handed weapon and grab corresponding fighter feats so you get some utility in your attacks. Only at the point where you can use both Bestial and Juggernaut mutagens would I switch to those. Too bad your saves suck now, eh?

Edit: To actually make a point here: I feel that to actually play a mediocre alchemist it requires far more effort than to to play a good any other class. The class has quite a few trap options, options which are plain worse than other classes and sometimes even glaring bugs (try holding 3 alchemic items in your hands... or your one hand if you use any weapon at all). To me it seems like they gave the alchemist a once over after the playtest and whoever was supposed to check if the class made sense at that point just dropped the ball. Sorry to say that the alchemist is the only, clearly exceptional class in this book and not in a good way.

Not sure if it's stupid or functional but I kind of want to do something with an alchemical crossbow on an Alchemist. Maybe MC ranger for crossbow ace? Not really sure.

It really depends if you want to mostly rely on the alchemical crossbow. If you do, you're better off with going Ranger MC Alchemist, as you dont particularly need better versions of bombs (alchemical crossbow only takes lessers bombs) and ranger gets better proficiency, feats (hell, quite a few alchemist feats are just downgraded ranger feats) and generally damage due to hunters edge (I'd take precision for xbow usage)