Your players shop between sessions.

No. Auto-Populate picks from public library items and your own items and suggests them; nothing goes on the shelf until you accept.

Set a restock interval in days. When it comes due, the shop re-rolls its stock from your list, replacing the previous stock. Scheduled restocks appear when you next load the shop and send no notification. The GM can also restock by hand, which updates open shop views and notifies the party.

Yes. Any revealed shop buys items other than currency at its own offer rate, while it has the funds.

The first purchase gets it. The second player sees that the stock changed, and nobody is charged for an item they did not get.

Yes, for the party: /shop buys and sells one item at a time through the party’s inventory and funds.

Nothing you've made goes away. You'll see an upgrade prompt when you try to go past a limit, like opening a second shop.