The vault is a pool kept apart from storage: your industry can't produce from it and trade can't export it. You park things like gold there, and its total value reads as cover against your debt.
Turn on automatic accumulation and set a target per item; whatever sits above your storage target is pulled into the vault before it gets exported. Capacity scales with your daily output, not your treasury.
Only reserve assets can be bought into the vault: gold, silver, gemstones and Bitcoin. You buy into the vault from the market or sell out of it; the effect never lands the same day, it works into the price over a week or two. You buy a little above and sell a little below the market, so churning trades won't print money.
It isn't. A lender you fail to pay can seize from the vault too, arrears freeze it, and bankruptcy liquidates it. A bloc you belong to can also require you to hold its quota there.
Every country keeps its books in its own currency; amounts on screen are in yours (you can switch to $ in Settings). The interest rate curbs inflation and strengthens the currency, but slows construction and raises interest on debt. If your currency loses real value, exports and tourism become competitive while imports and foreign debt get dearer. If the real rate slips into the negative, cash in the treasury erodes. In a union like the euro, or if you use another country's money, you don't set the rate; under a peg the rate follows the anchor's.
In the short term, yes: money flows into the treasury every day. The bill comes later: inflation climbs, your currency loses real value and approval erodes as purchasing power falls; if the real rate slips negative, cash in the treasury and domestic debt erode too. You can curb inflation by raising the rate, but that slows construction. In a currency union, or using another country's money, you can't print.