Do x-year treasuries become y-year treasuries (x-y) years later?

Yes, these are called "off-the-run" treasuries. Since they are not sold by the treasury directly but sold in secondary markets, they are less liquid than "on-the-run" treasuries and are cheaper (have a slightly higher yield) that the more liquid treasuries sold at auction.

Also, since a "10-year treasury" is just a treasury bond that matures in 10 years, and the maturity date is fixed, in a sense they do become "5-year treasuries" after 5 years, but I do not know how brokers actually list them (e.g. they may list them as "10-year 5% notes maturing YYYY-MM-DD" and you do the mental math to determine the time remaining). The key metric when pricing these bonds is the time-to-maturity, not the original tenor of the note.

I doubt anybody in the world is looking to buy a treasury bill expiring in 4 days.

If someone has a 30-year note that is expiring in 4 days and is willing to sell it for a lower price than current 4-day treasury bill (which doesn't exist, but say it did for the sake of argument) why wouldn't you buy it? You'd get a better yield than if you bought one from the treasury directly.

I can't speak to the actual market for 4-day treasuries, but anything is for sale for a price...

The price for this bond must be essentially 0 because of no demand

That's not how bonds are priced. Why would someone sell a $1,000 bond for zero when they're guaranteed to get $1,000 in 4 days? The "clean" price of these bonds should be very slightly below par. If the bond pays a coupon, then you'd also pay whatever interest has accrued since the last coupon was paid (called the "dirty" price). So for a $1,000 bond with a 5% coupon ($25 every 6 months), you might end up paying $1,024.50 for a bond that will give you $1,025.00 in 4 days.