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.
An 'Alchemical Item' counts as any mundane item that can be created with Craft (Alchemy). Potions count as Potions, and even though they can be made using Craft (Alchemy) they are never referred specifically as an 'Alchemical Item' anywhere in the rulebooks.
Although, Alchemist classes making potions at half-speed is not game breaking (it even makes sense!), so as a GM I would allow that as a house rule.
Best Answer
Creating Extracts requires only a minute of free time, a Formula Book, and an Alchemists Kit (or other source of suitable raw materials if your DM is kind). No special environment is required, and the ability to brew extracts to fill unused slots on-the-fly in the field is considered a given for the class, as written. Note that an Alchemists Kit is not an Alchemists Lab, and may not necessarily be assumed to be sufficient for doing things like creating Potions, Poisons, or mundane crafted Alchemical items. It is instead, roughly an Alchemists equivalent to a Wizards Spell Component pouch:
As with a Spell Component Pouch, it can be assumed to be kept restocked by an Alchemist under normal circumstances without maintenance or significant cost. Those extracts with specifically delineated and named material components will need to have their requisites secured and stocked separately, as is usual and customary.