Alchemical weapons are both alchemical items and weapons
Alchemical weapons are just that, weapons. It’s right there in the name, and nothing in their description says “despite the name, they aren’t actually weapons.” Unless something says otherwise, rules-as-written, anything you can do with a weapon can be done with alchemical weapons. That includes drawing them, and selecting them for feats which require you to select a weapon.
They are, for exactly the same reasons, also alchemical items, and anything that applies to alchemical items also applies to them. Specific trumps general, so, when Quick Draw says you can draw weapons faster, and then later specifies that you cannot use it to draw alchemical items faster, the ban on drawing alchemical items would also block drawing alchemical weapons faster with the feat (but you could still draw them normally for weapons).
For drawing, this is probably moot: unlike Quick Draw, the regular draw action can apply to “weapon-like” non-weapons
Where Quick Draw explicitly bars various magic gear like wands and alchemical items from its benefits, the regular move action to draw an item does quite the opposite. It actually explicitly applies to non-weapons:
This action also applies to weapon-like objects carried in easy reach, such as wands.
So since alchemical weapons are weapons, tautologically they are also weapon-like, and so the rule even-more-certainly applies to them. The free-action drawing at BAB +1 does not repeat the weapon-like rule, but in my view, “this action” here refers to the idea of drawing a weapon, not to the move action specifically, and so this rule would still apply. If you don’t buy that, well, alchemical weapons, at least, are still weapons, but I see no reason why drawing a wand from your belt should be harder than drawing a sword.
About the Quick Draw feat
The ban on various “magic” gear from Quick Draw’s benefits is a Pathfinder addition to the feat, relative to the version found in D&D 3.5e. That ban included alchemical items. If alchemical weapons weren’t weapons (or weapon-like) and couldn’t be drawn with the version of Quick Draw that lacked the ban on alchemical items, that line wouldn’t be necessary. Redundancy isn’t completely implausible, of course, but it’s hard to argue that Quick Draw’s explicit ban—which would be unnecessary if the answer were “no”—is evidence that the answer should be no.
On the subject of Quick Draw, it is worth noting that there was absolutely no balance concern here; there never had been one in 3.5e and Paizo’s stated reason for the change was because “it didn’t look right.” See this Q&A for more details. I personally recommend ignoring this change entirely; I think it damages the game by making the rogue et al. less able to pull out the right tool for the job at the right time, when really that’s supposed to be a big part of their whole schtick.
About the underground chemist’s chemical weapons
Well after the CRB was published, Paizo published the Advanced Class Guide which included the underground chemist archetype for the rogue class. It gets a chemical weapons feature that grants, as part of its benefit, the ability to draw alchemical items as if they were weapons. As an archetype ability for a particular class in a supplement, generalizing outward from it to the rest of the system would have been a no-go, at least RAW-wise, but you could make an argument that it indicates that Paizo considered alchemical items to be not-weapons as far as drawing was concerned.
Except that the feature says you can draw “alchemical items,” and not all alchemical items are alchemical weapons. The un-weapon-like ones would not be eligible for drawing as weapons under the default rules—thus the benefit of the chemical weapons feature, which allows the underground chemist to do so.
About “prepare to throw a splash weapon”
The actions in combat section of the rules lists various actions, and under full-round actions, it includes “prepare to throw a splash weapon.”
This is, unfortunately, not backed up by any text anywhere outside of that entry. We have an entire Q&A dedicated to this conspicuous absence. As a result, we have no idea what it means to “prepare” a splash weapon, when or why you need to do it, how long the weapon remains “prepared” once it is done, whether or not preparing includes drawing the splash weapon, whether or not it includes throwing the splash weapon. This is a full-round action that we know nothing about, as far as I can tell. I can find no clarification of it anywhere—and that entry comes from D&D 3.5e, which also had zero information about it.
Ultimately, from a RAW perspective, rules text is generally seen as primary, while tables are secondary. The D&D 3.5e errata rules explicitly defined this; Pathfinder doesn’t have as detailed rules about conflicts, but it’s a reasonable position to take there, too, particularly since this table entry is a 3.5e relic anyway. Since the rules text never mentions any need to prepare a splash weapon, you never need to take this full-round action, and if you did it would do nothing because splash weapons don’t need to be prepared.
Anyway, it does not provide any meaningful elucidation on our question here.