Things you can do, by the book: Everything you listed except scribe a new spell.
Source: PHB pg. 114
That actually requires you to write down the spell, and has a material cost associated with it that is usually associated with special inks and gems. Yes, you can recall it from memory and write it into your book, upon which you would have it memorized for a month. However the spell itself may require a specific rune written in a specific ink. Knowing that, and having written it down that way are two different things. Obviously this applies to material component cost spells only.
Since the Wizard can already replace a destroyed spell book with the spells he has prepared for the day, it's reasonable to extrapolate that with photographic memory (as this feat entails) you could replace the entire book. After all, the spells don't disappear from your mind when you cast them, so this feat should serve as a good safeguard against losing your spellbook.
Of note: You're going to want to review your book monthly and update it with new spells in order to continuously preserve this.
Of secondary note: I would even allow a Wizard with this feat who passed a successful Arcana check to experiment with and replicate a spell cast by somebody he was watching. Personal DC levels would be, DC +5 per component needed (Verbal, Somatic and Material), and double the result if the Wizard isn't proficient in Arcana. But that's a house rule thing because this is a clever use of a versatile feat that effectively duplicates the spell book.
But bottom line is: Nothing under Wizard or the Spell Book sections in the PHB prevent you from recalling everything in your book from memory. The pertinent line is here:
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: at least 1 minute per spell level for each spell on your list.
Since Keen Mind already ensures you have the spells memorized, there`s no reason you would need to open the book in order to review them.
No you cannot transfer spells prepared as a Cleric into your spellbook
Just above the text that you reference, the sidebar in the PHB explains
When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare
the time to decipher and copy it.
The fact that the book explicitly says that it contains wizard spells is the killer here. Spells that you prepare as a cleric are not wizard spells. They are cleric spells, even if they are on the same spell list.
We know this because of the rules for multiclass spellcasters (PHB pg. 164)
Each spell you know and prepare is associated with one of your classes, and you use the spellcasting ability of that class when you cast the spell. Similarly, a spellcasting focus, such as a holy symbol, can be used only for the spells from the class associated with that focus.
From this, we establish that your prepared spells as a cleric are "Cleric Spells" in that they are prepared as a cleric, and cast as a cleric; not as a wizard. And your prepared spells from your spellbook are "wizard spells" for a similar reason. Even if the two appear on the same spell list, each spell you have prepared is associated only with one of your classes.
The PHB then goes onto explain that when copying spells from spellbook to 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.
If copying from book to book is the same as adding new spells except faster, then we know that you can only transfer over wizard spells since you can only copy new wizard spells into the spellbook.
Finally, As you have pointed out:
you can use the same procedure to transcribe the spells that you have prepared into a new spellbook
The procedure in question is the same as copying over a new spell, or a spell from one book to another. Both of these processes require wizard spells to work. Thus, even though copying a spell down from memory doesn't specifically add any new restriction, it doesn't explicitly lift the general limitation on the procedure; the precondition that the spell in question is a wizard spell.
Best Answer
Yes, you can cast spells without your spellbook. And yes, your list of prepared spells will stay prepared until you prepare a new list. In the Preparing and Casting spells section, only preparing a new list of spells actually requires your spellbook. Casting just requires spell slots and a prepared list of spells, and regaining spell slots just requires a long rest.
There are some other things you can't do without your spellbook, however:
A suggestion for you: The game assumes that a Wizard has a spellbook, and a lot of the Wizard rules don't quite work without one. If you lose your spellbook, get a new one as soon as possible. For the pittance of 50 gp, you can make the rules work properly again. Even if you still don't change your list of prepared spells, it'll be worth it.