Okay - Lots of questions means lots of sources. So lets first look at the main spell, simulacrum.
... It appears to be the same as the original, but it has half the creature's hit point maximum and is formed without any equipment. Otherwise, the illusion uses all the statistics of the creature it duplicates, except that it is a construct.
... The simulacrum lacks the ability to learn or become more powerful, so it never increases its level or other abilities, nor can it regain expended spell slots.
So the key points are that it's a weaker duplicate of the wizard, and cannot learn.
Now lets see what it takes for a wizard, to cast a spell. Under the general rules:
Before a spellcaster can use a spell, he or she must have the spell firmly fixed in mind, or must have access to the spell in a magic item. ... Other spellcasters, such as clerics and wizards, undergo a process of preparing spells. This process varies for different classes, as detailed in their descriptions.
So wizards prepare spells, and we'll look at that in a moment. First, take a look at ritual spells since that will come up shortly.
To cast a spell as a ritual, a spellcaster must have a feature that grants the ability to do so. ... The caster must also have the spell prepared or on his or her list of spells known, unless the character's ritual feature specifies otherwise, as the wizard's does.
So wizards can cast ritual spells, but they do it differently than others. So let's see what it says about wizards casting spells, both normal and ritual.
You can change your list of prepared spells when you finish a long rest. Preparing a new list of wizard spells requires time spent studying your spellbook and memorizing the incantations and gestures you must make to cast the spell.
You can cast a wizard spell as a ritual if that spell has the ritual tag and you have the spell in your spellbook. You don’t need to have the spell prepared.
So those are the wrinkles to all of this. Now lets look at the questions again:
1. Prepare new spells after a long rest?
For wizards, to prepare a new spell, requires studying and memorization. This reads to me as "learning" which a simulacrum cannot do. So there is no way for a simulacrum to learn a new spell, only use the spells in its head at time of creation.
2. Cast spells as rituals?
There is nothing stopping it from reading directions from the book--assuming a simulacrum can read your notes. But since you wrote the notes in the first place, and the spell description says that it is the same as the original and has all the same stats, I would say that reading your own handwriting would qualify. And as a bonus, since rituals don't use spell slots, you can cast them repeatedly as a simulacrum cannot regain spell slots.
3 .Copy new spells into the book?
4. Copy their currently prepared spells into a new book that is then usable by the original wizard?
These I am lumping into the same category. The spell states that the simulacrum "cannot become more powerful". Adding new spells to a spellbook, whether into the existing spellbook or in creating a new one, would fall into the realm of "more powerful". However, since the simulacrum cannot actually use the new spell (can't learn it), its up to the DM to decide if that would make the simulacrum more powerful or not.
To the other point, if a simulacrum is tasked with copying a spellbook, they would use the same notation as the original. In fact the PHB states:
Replacing the Book.
You can copy a spell from your own spellbook into another book—for example, if you want to make a backup copy of your spellbook. This is just like copying a new spell into your spellbook, but faster and easier, since you understand your own notation and already know how to cast the spell."
The simulacrum would copy the spellbook using the same notation as the original so they would not be able to create a spellbook that the original could not read.
So the short answer is, the simulacrum can cast rituals so long as they have the spellbook handy. They cannot change memorized spells ever. And copying spells is DM interpretation.
