I'll help as much as I can. Here's some of the things you see hanging in a barn:
Pulleys or sometime simply ropes over a beam to lift
Hay hooks- We've got 4-5 and they are all at least a little different. Easily made by a blacksmith they show a lot of variation Here's some pics I found.
Chain for heavy moving like logs and small sheds built on runners.
Pitchforks (3 and/or 4 tine), Shovels, spades, rake, hoe, post hole digger
Axes and hatchets, you use a hatchet quite a bit for things like chopping holes in the rain barrels and troughs and busting gounds to feed cows.
Scythes and corn knives, We've got several corn knives and 2 machettes. Corn knives have an edge but no point.
Pails- for milk, grain, water, anything
Bushel baskets- emptys mostly, full ones would have been in the root celler
A scoop for feed
A corn sheller and a grinder
Crates and barrels- Barrels were mostly outside to catch water but one or two inside to hold grain that had been shelled, cracked,... processed in some way. Crates are a big catch all. They were kept in case they were needed and always end up with stuff in them, frequently gunny sacks, saddle bags, whatever has no other place at the moment or that just gets left there. A crate with sacks in it or the lid covering part of it or both is a prime spot for a cat to have kittens. My old mama cat had her 2 yr olds in a crate of sacks in the barn. We also have a pianobox used to store grain!
Carpenter's tools, two man saw, general purpose ladder, wash tubs that have been replaced with newer ones in the house, brace and bit, kegs of nails, odds and ends lumber or sheet metal, windows
A lantern or 2, kept well clear of the hay and stock.
Meat hooks, scalding and rendering kettles
Chicken crates, stock tack, livestock bells, sheep shears, horse shoes, any horse or ox drawn implement or cart, sleds, bicycles, just about anything that needed inside that didn't go in the house.
Garages and garden sheds come later as farming expands or not at all it the room is lacking. The house and the barn come together. Stuff may get moved from the barn to another building but not the other way unless the building is being moved, repaired or torn down. I've seen sheds and covered feeders built in sections inside in the winter and assembled outside when weather let up. Remember storing is a matter of creativity and thinking in 3-D. A ladder can be stored in the air, laying between two crossbeams. Same with light implements. Things get re-purposed with time. An old, leaky wash tub will still feed sheep. Old horse shoes become hangers. Old, worn or damaged tack gets kept for parts. Same with equipment. A low (40") movable "gate" or partition (simply a wooden frame with board across it and maybe a diagonal brace) used to separate small, sick, or young stock from the rest on occassion, like when a ewe has lambs.
What kind of hitch will your cart have? A tongue or staves or have you not decided or know yet. How the cart is made will dictate what kind of tack is used. Staves are for one animal. A tongue(double-tree) is for two. You either will need a double set up or single, for oxen or horses depending.
Tell me what you see and I'll help from there.