[RPG] Does an alchemist need make a Spellcraft check when he wants to copy a formula

For feats, most characters get a feat at first level, and then another feat at every odd-numbered level. Humans get an additional feat at first level. So, your example 5th level Elf will have 3 feats available. If you have a WOW background, think of feats as talent points - they let you customise your character a little more every couple of levels. This is spelled out in the table on p30 of the Core rulebook.

The bottom of p27 of the Advanced Player's Guide (APG) outlines how many formulae are in your book by default - at first level, you get 2 1st level, plus an extra first level for each Int modifier. At each level after first, you get another 2 formulae of any level you create. So, by 5th level, you would have 10 plus Int modifier formulae, of which up to 4 could be 2nd level formulae.

You can also use bonus elixirs each day based on your Intelligence, so you'll likely get an extra one of each level, but check the bonus spells table in the Core rulebook to be sure (not sure what your Int is).

Elixirs are totally separate to bombs, even though the rulebook confuses things by mentioning them both in the 'Alchemy' rule. The Bomb rule on p28 of the APG, halfway through first paragraph, says you can use a number of bombs per day equal to class level + Intelligence modifier.

The Brew Potion feat allows you to create a potion version of any spell in your formuale book. Think of it as making a lasting version of an extract. This is described on p551 of the Core Rulebook. You can also make alchemical items (i.e. Alchemist's Fire, acid etc), and the rules for this are covered in the Skills chapter, p.91. The items you can actually make are in the equipment chapters of the various rulebooks.

Good luck, and have fun with the alchemist.

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
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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
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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.