How do the formulas for alchemical items from the basic crafting book interact with the alchemical formulas from the Alchemist class

From the SRD (emphasis mine):

Although the alchemist doesn’t actually cast spells, he does have a formulae list that determines what extracts he can create
...
An extract is “cast” by drinking it, as if imbibing a potion—the effects of an extract exactly duplicate the spell upon which its formula is based
...
An alchemist can also add formulae to his book just like a wizard adds spells to his spellbook, using the same costs and time requirements. An alchemist can study a wizard’s spellbook to learn any formula that is equivalent to a spell the spellbook contains. A wizard, however, cannot learn spells from a formula book.

Alchemists cannot learn spells, they learn formulae, that are based on actual spells, but are not spells. Thus, if a wizard’s spellbook contains a spell that is the equivalent of a existing formula, you can study the spellbook to add that formula to your own formula book (essentially recreating the formula from the spell). However, the wizard cannot do the opposite.

Also, as a side comment, since alchemist extracts work like potions that must be ingested, makes sense that most formulae mimic spells that are either of personal range, buffs or healing, due that method of application. Attack and debuff spells could not be properly applied with that method (just imagine the consecuences of pouring a fireball down your throat), so it would not make sense that the alchemist could learn any wizard spell.

Can an Alchemist copy formulae off of a divine scroll?

Yes, an Alchemist can copy a formula from a divine scroll if it appears on their formula list. An Alchemist can copy spells into a formula book just like a Wizard copies spells into their spellbook. Therefore, if a Wizard can copy a divine scroll, an Alchemist would be able to as well. Because divine writing can be deciphered like arcane writing, and the rules for copying spells from scrolls does not discriminate between divine and arcane magic (it just says "a magic scroll"), a Wizard would be able to copy a divine scroll into their spellbook if it appears on their spell list. Alchemists can do the same, but they do not have to decipher the writing first, as long as the scroll's spell appears on their formula list.

Can an Alchemist copy formulae from another Alchemist's formula book?

There is no RAW answer. However, one could argue that an Alchemist's ability to copy spells into formula book as a Wizard copies spells into their spellbook means they can duplicate an existing formula book like a Wizard can duplicate a spellbook. Because of that, an Alchemist would have to be able to copy spells between formula books. Additionally, Ultimate Magic includes a formula book in its example spellbooks, which would imply someone would be able to copy from it. Using this evidence, an Alchemist would be able to copy formulae from another Alchemist's formula book.