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When you say you can't add any formulae, could you be more specific about what's preventing you from doing that? I just tested on one of my own alchemist characters - On the Crafting Formulas tab, I went to Alchemical Formulas, and scanned the list of formulas to purchase, and bombs were in there.
Is the issue with bombs connected to this - that none of your original formulas were bombs and you haven't been able to add new bomb formulas, or was that a separate issue?
Neither of those feats grant additional formulas, as far as I know. Expert alchemy increases your alchemy level so you can craft higher level alchemical items.
Your screenshots are just showing your class page, with class feats and free formulas granted by your dedication. Did you look under Build > Equipment > Crafting Formulas > Alchemical Item Formulas? You have to go there to buy additional formulas.
Glad that helped! Yeah, it's a little different than spellbooks, where you enter both free and purchased spells. The list on the class page is just granted formulas, and they also then display in your equipment list.
One warning—you can delete any formula from the equipment section, including the free formulas you selected as part of your class feature. You'll get a warning that it was "added from another table" and ask if you want to proceed. If you do, it will delete it from your crafting formulas under equipment AND from the formulas chosen on your class page.
No, don't add anything for free. I haven't run an alchemist so am not 100% on how it works, though. You will define batches from your available formulas. Those will use your daily allotment of reagents. You'll click the + icon next to each batch to create it as part of your daily preparations—that will add it to your alchemical equipment under infused items created by you.
I don't know if those automatically go away after a night's rest (it doesn't look like it when I try and test it), so if you change up your daily batch setup you may need to delete some infused items and refresh the batch the next day. Hopefully someone from Lone Wolf or who plays an alchemist can give you better details.
If you're an alchemist or some archetype that allows you to create temporary items, there will be a table of batches to prepare (the exact terminology may vary whether it's alchemical reagents, snares, temporary talismans, etc). There, you add the list of items you want to prepare, and choose how many batches of each to create. Then, the left-most button will create an infused stack of that item, with the correct quantity. For a new day, we assume that most alchemists won't vary their choices much, so we don't just clear the whole set - for each item where you've used some, go to the same button that created the infused stack, press it again (there will be a different icon, but in the same column), and it will reset that stack back to an un-used state with the original quantity.