[RPG] Can an alchemist create extracts with metamagic feats

Activating a metamagic rod gives you “the ability to use the given feat,” which would then be subject to the rules for the use of metamagic feats—including the one you quote forbidding their use with spells cast out of items. So no, you could not do this. Generally speaking, the only way to cast a metamagic’d spell from an item is to create an item specifically containing that spell already metamagic’d (and paying for the increase in spell level, accordingly).

However, the ring of transcendant spells is worded differently. It does not specify that it allows you to use a metamagic feat—in fact, it doesn’t specify that it’s applying a metamagic effect at all. It just modifies spells you cast, and you do cast spells from wands and scrolls. Thus, it could be applied spells cast from magic items.

Unless I'm misreading the description, I think the item you're looking for is the Caster's Tattoo.

A caster’s tattoo is infused with potential magical energy that aids in casting spells that normally utilize verbal and somatic components. Once per day, a spellcaster may activate her caster’s tattoo as a swift action to automatically enhance the next spell she casts in that round with both the Still Spell and Silent Spell feats. When she casts the spell, the caster’s tattoo flares with light for a moment, then grows faint and faded—it replenishes its energy in 24 hours, after which it becomes bold and colorful and can be used again. Required material components or foci must be in the creature’s possession to cast the spell (and are consumed as normal), but need not be held when the spell is cast.

Caster’s tattoos are available in lesser, standard, and greater versions, allowing the modified casting of a spell of up to 3rd level, 6th level, or 9th level respectively.

Its only once per day, but nothing stops you from getting multiples if you're dedicated to having this capability.