You would not keep background features

Since features granted by background are not mental abilities or class features, you do not keep them. Also, generally ability score increases and proficiencies are kept separately then standard features so you could not keep proficiencies even if they were gained by a class feature (but not if you "have" proficiency). The wording matters.

Some insight from Jeremy Crawford, lead designer:

hp, which are replaced by the beast' s hp while using wild shape.

Since the qualities that are one-time benefits are replaced (anything that is a permanent buff to an attribute like HP or Strength for example), you don't get to keep them even though they are class benefits.

The Simulacrum cannot "replace" a spellbook they never had

It's important to keep in mind that a Simulacrum copy of you is not you. This manifests in many ways (if you have a familiar, they cannot dismiss it: if you are concentrating on a spell, they cannot end it), but one in particular stands out: your equipment is not theirs.

This is not only logical, it's part of the spell's description, which states that the Simulacrum copy has most of your game statistics, but like you said there are a few major exceptions. One being (PHB, p. 276):

It ... is formed without any equipment.

This doesn't just mean that they appear without a copy of your equipment: it means that your equipment is not theirs. One helpful example of this is that if you are attuned to a magical item, the Simulacrum is not similarly attuned to it as well. Similarly, if you created a spellbook, it isn't their spellbook too.

Now, you're completely correct that the Simulacrum could create their own Wizardly Quill. But when they attempted to "replace" your spellbook, they'd run into the following problem (TCE, p. 78, bold added):

If necessary, you can replace the book over the course of a short rest by using your Wizardly Quill to write arcane sigils in a blank book or a magic spellbook to which you're attuned. At the end of the rest, your spellbook's consciousness is summoned into the new book, which the consciousness transforms into your spellbook, along with all its spells.

Now, the first part of this description could apply, so it looks at first like this might actually work. But when you get to the bold section, it falls apart. The Simulacrum can't summon "their spellbook's consciousness" or fill a book with "their spellbook's... spells" because they don't have a spellbook: they never did, because they were created without equipment, and they are not you. Thus, there is nothing (neither consciousness nor spells) to transfer into this blank spellbook when they attempt this process.

One might argue that the Awakened Spellbook isn't equipment at all: that it is a creature. Now, that may be (the rules are unclear on this), but in that case it will run afoul of the text of the Simulacrum spell (PHB, p. 276, bold added):

You shape an illusory duplicate of one beast or humanoid that is within range for the entire casting time of the spell.

The Awakened Spellbook probably isn't a beast or humanoid, but that's beside the point: the important detail is that if we are arguing that it is a creature, it is a separate creature from its creator (who is not an awakened spellbook). The two beings may be strongly magically linked, (like a wizard and their famliar), but that is different from being the same creature. When the wizard is copied by the simulacrum spell, they are the only creature the spell copies.

The upshot (and how to get around it)

Thus, whether we consider the book to be equipment or a creature, the Simulacrum copy can't use the Awakened Spellbook method of "replacing" their spellbook when they are first created, as it only works if they have (or ever had) an Awakened Spellbook already (which they don't and didn't).

That being said, they could buy a new spellbook, and then awaken this new blank spellbook, thus being able to use all their class features again. If they are ok with this spellbook being empty (containing no spells, but being able to be used for the Manifest Mind, and other Order of Scribes features), then this would cost just 50 gp, according to the rules on adventuring gear (PHB, p. 150).

Now, if they wanted to have all the spells from your spellbook in theirs, they would have to physically copy them in. This would be considerably more expensive (much like it would be expensive for a simulacrum of a fighter to get outfitted with its own armor and weapons), but they won't be entirely out of luck there either: their Wizardly Quill will let them do so at only 2 minutes per spell level, (TCoE, p. 77), and they can use your spellbook as a guide to copy the spells, debatably using the rules a wizard uses for copying their own spellbook (PHB, p. 144) (10 gp per spell level: credit to BlivetWidget for pointing that out) since they also "understand... your notation" (ibid). But they won't be able to do so automatically as part of a short rest. Oh, and you'll need to provide them with some gold to spend on the rare materials they'll need to copy the spells. Because of course, they don't have any equipment.