Contest should always happen for achieving control over something. in the example you talk about grappling and pushing is freedom of movement or space. Another example could be for example someone blocking a door while someone is trying to open it.

Sometimes one character’s or monster’s efforts are directly opposed to another’s....This situation also applies when one of them is trying to prevent the other one from accomplishing a goal—for example, when a monster tries to force open a door that an adventurer is holding closed. (Basic D&D p58)

It could never be a direct attack so for example it could never be

• setting someone on fire with a lamp (it's an attack , you crash it on her)
• blinding someone with sand ( that's also an attack )
• flipping a table in the face of someone ( attack )
• dragging the carpet under the feet of someone ( attack )

Sand , tables, carpets are all improvised weapons that could do no damage but they could cause effects or trigger an opposed roll. At least is how I see it.

Now situational bonuses or penalties do apply, for example a lamp can be used as a flail but it's not one, so, a penalty to attack and damage should apply. Sand is not designed as a weapon so it doesn't count as a throwing dagger. Each case is completely different from the other, you cant forecast what the players might think in their desperation or just for having fun and it's not worthing to spent time designing improvised weapons. Just tackle it as it happens, agree with your player's at the spot with a formula that will apply for that moment only. What matters is for the improvised attack to contribute to fun, by being just as hard and rewarding to avoid letting your players using this solution as a loophole especially if looks temptingly effective. Make things interesting by raising the stakes, increase effect and difficulty (made up some reason) or add a backfire case if the attack fails.

I don't see any reason though why not using blindness as the condition when for example you wrestle with someone to pick wis eyes out or fight with someone to extinguish the only source of light in the room. But in this case you actually fight over the torch, Blindness is a side affect because there is no light in the room anymore.

Good examples of contests are

• Fighting over a light source.
• Wrestling to grab the one ring from the floor (Golum vs Frodo)
• Trying to Swim while someone is dragging you down. (You both truing to reach the surface while trying to kill each other at the same time)
• Trying to disarm someone by means of grappling. (You both trying to hold on the weapon)
• Playing Pink Pong or Tennis :)
• Drinking Contest

In any case, Contest is a hack to the rules in order to cover some unexpected cases when it's not covered by the existing rules and usually the effect is more of a strangle towards something. How exactly you are going to deliver that, is what you have to improvise.

Again as with the improvised attacks, there is no rule that tells you that someone is not going to wrestle to blindness, to castration or to death. (All that actually happened in ancient olympics btw)

Anything that involves strangle while doing the same thing or opposing over the same thing is a valid Contest. Anything that is actually like "I'll grab the carcass of the giant frog and I will brake your head" is an improvised weapon attack.

Contests tent to be more abstract and Improvised attacks more specific. Both are abstract rules for actions that are not ruled by the book and it doesn't really worth in most cases to worry about. by experience you make rules of the thumb and you keep the game going.

To conclude You can have whatever effect you like with your improvised action as long as it makes sense.

• Use improvised action and Contests for more generic actions.
• Use improvised weapon when you have an improvised implement.
• Make things interesting cooking the difficulty/effect relation

P.S. SevenSidedDice mentioned a few examples and over that we disagreed. The first one is if grappling is an attack. Yes it is and instead of unarmed attack vs AC : resolves to dmg it's actually STR vs STR : resolve to grappled.

Trying to blind someone by throwing sand though should be an attack, using sand as an improvised weapon. Because the implement of the action is in your hand.

Using the cloak of the opponent to blind him... well the item belongs to the opponent, you have to take control of it first, then you have to move it to her face to cause a blinding temporary effect. There are many ways of doing this but two of the could be. Either you simplify it with an improvised action DEX vs DEX. Or spit it to two actions, a sleight of hand to grab the cloak and an improvised weapon to actually blind the opponent with her own cloak.

So, while improvising actions, you can choose between using improvised weapons or improvised actions or mixing both depending on the context.

Interacting with an unwilling targets items, I would considered an attack. but again whatever makes sense depending on the context.

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.