If I were to price a bond on one of its coupon payment days, does that day's coupon payment gets added to the cashflows, if so, do we just discount that by 1 (same day)? ie, C1*df1 + C2*df2 + ... Should C1 be that day's coupon payment, or the next period's coupon payment?
There is the concept of ex-div on stocks and bonds. This is short for ex-dividend (or ex-coupon). If you transact a bond or a stock on an ex-div date then, as the investor, you will not receive the dividend or the coupon which is imminently due to be paid, usually within a few a days or weeks. Instead, as an investor, the first coupon you will receive is the one after that.
This is why you see stock prices fall on ex-div dates since they reflect the value of the different cashflows for investors. It is also apparent in the dirty price of bonds, however most bond price charts are clean prices and already exclude the accrued interest so this is not noticeable on those charts.