Learn English – I thought they were/are 9us(size)

To suppose that something is true means to think that it is true, but to freely acknowledge that it may just as well be false. To assume that something is true means to act as if it were true, but with no implication that there is proof that it is. To presume that something is true is to base an assumption of truth on some sort of probability or evidence.

So, "I suppose you are using Internet Explorer" is a slightly weaker assertion than "I assume you are using Internet Explorer." For an example of presume: "One may presume that he is using Internet Explorer, because he is using Windows and is a computer novice."

These are not hard-and-fast rules; as you have seen the meanings overlap to some degree. This is especially true of suppose and assume.

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.