Learn English – “to keep something as a memory” or “to keep something as a memento”

I am a native UK English speaker:

Repository - central location to store something for the sake of safety/preservation. An example of this would be the storing of code e.g. github where you want the code to be safe. Another example would be the storing hazardous materials e.g. nuclear waste.

Depot - Although this is described as a synonym of warehouse/storehouse, I only hear this used to describe places where people store large equipment while not in use - for example, storing buses overnight when not in use or storing buses in need of repair. The equipment is generally used off-site.

Depository - A place to store things for safe-keeping - particularly valuable items. There is a security aspect to this. An example would be a library, museum or safety deposit box at a bank.

Warehouse - generally means large building you store things. Usually it is used to descibe a large building to store finished goods in, think the definition of 'wares' (manufactured goods). A large department store such as IKEA would have a warehouse to store it's large number of stock items. A small retailer e.g. a small book store or supermarket would likely use a 'stock room' for local storage of goods sold in the store and have a number of more central warehouses that store large amounts of stock ready for distribution to individual stores' stock rooms. Warehouse is a very generic term and can also just mean large building you store things in bulk in.

Storehouse - A place you store things, very general refers to a building usually. Depositories, repositories or warehouses could all be described as more specific types of store houses. I don't use this word a whole lot as it's pretty general, you may as well say 'X is stored offsite' or 'X is in storage'

Question 1: The sales assistant would say:

"I am not sure. Let me check the stockroom"

If she checks the stockroom and finds no shoes currently in store she might say:

" Sorry, we currently have no shoes in stock in store - I could check our warehouse for you?"

You agree and she checks the store database to see if they have any shoes in the central warehouse. If there are, she might order them for you to be delivered to the store.

Question 2: I would say for finished products - 'warehouse' works in most situations, I very rarely hear people use the word 'storehouse'. Many raw materials are kept in storehouses with specific names e.g. grain is kept in a silo or granary. A silo is a large tower that's filled with grain/silage for storage. However, if you said the grain was stored in a storehouse or warehouse people would know what you meant and it would sound fine.

There is ambiguity in your isolated phrase

A contacted B with a contract

the possibilities are

Is A merely contacting B and one of them has a contract?
Does A have the contract?
Does B have the contract?
Does A have the contract with the intent of giving it to B after contacting B?

By the Law of Proximity(1)

A contacted B with a contract (B has the contract)
With a contract, A contacted B (A has the contract)

Because of context from your passage

A (has a contract and has) contacted B with (said) contract (to give to B)

since A is B's handler and handlers do not phone-in for updates, they wait to be notified (minimal communication)

In your example question

A visited / approached / ran to / notified / engaged B with a contract.

Let's try this

John visited Jane with flowers
John visited Jane with Sally

I think most people would interpret it as the flowers and Sally were with John

John approached Jane with Sally
John ran to Jane with Sally
John notified Jane with Sally

would also be interpreted as Sally being with John

change with to and and any ambiguity disappears

John approached Jane and Sally
John ran to Jane and Sally
John notified Jane and Sally

however

John saw Jane with Sally

would usually be interpreted as Jane and Sally were together.

(1) this may not be a real law, but is usually understood