When Did The Banks Stop Keeping Track Of Half Cents?

We can make some basic assumptions.

1) Most banks would only have a 10-20 year charter. This would limit the tracking of Half Cents to a maximum of 1877 (based on a bank starting in 1857) as in 1858 banks would no longer be obligated to take them.
2) A 1/2 cent in 1857 barely had any value (and it's metal was worth more than the monetary value, kind of like today with people pulling the wheat and silver coins from circulation)
3) Half Cents only circulated in large eastern cities, they rarely circulated outside of cities.
4) (Conjecture) In 1858 banks probably stopped taking 1/2 cents as deposit, you may have needed 2 1/2 cents.
5) In 1858 any Half Cents deposited would have been sent for melting as they were no longer accepted and the metal was worth more than the coin.
6) The flying eagle and Indian Head cent were very popular with many people trading large cents for the white cents, hence many large cents and Half Cents were removed from circulation
7) During the civil war almost all coins were pulled from circulation.

Based on the above I would believe most banks stopped tracking Half Cents relatively soon after the release of the flying eagle/IHC. I would believe by 1857 country banks wouldn't even see Half Cents so they probably tracked in full cents. I would say by 1859/1860 majority of banks rounded up or down any Half Cent balances.

The only outliers would probably be banks near the Canadian border as large US cents circulated with the large Canada cents until 1920. If a bank for example was in Niagara Falls NY it might have tracked Half Cent balances until the late 1800's (like 1880-1890).

That's my best guess based on what I know.

The United Kingdom decimal halfpenny, introduced on February 15, 1971 and withdrawn on December 31, 1984, was never shown on bank statements and you could not put it on a cheque. You could certainly get them from banks and pay them into your account, but I think the main reason they were left off statements and cheques was because they would have caused problems with electronic account systens, which were just being introduced. I suspect this was probably the main reason the coin was demonetised - many European countries had coins of lower value in circulation after 1984. I had a French bank account around that time and could make cheques payable to the nearest centime (one centime was then worth about one-tenth of a British penny).

I was under the impression that most late-is sue Half Cents were shipped to Post Offices to use as change? I wouldn't be surprised to see a half-cent in a bank balance due to deposited scrip.

They were never obligated to take them. Half Cents were never legal tender until the mint act of 1965. So they were LONG out of circulation before they ever became legal tender.