Let's say I write a covered call and a buy separate put option. After a period of time, the call expires, leaving me with the underlying stock. Thereafter, the market price went down and I can exercise my put option. Will the underlying stock that I own before be use for the assignment?
Yes, if the call expires worthless, leaving you with stock. Then you can exercise your put when the stock goes below put strike price.
Owning 100 shares and selling a covered call to fund the purchase of a put is called a long stock collar. If the strikes are different, it is equivalent to a vertical spread. If they are of the same series, it's a conversion.
The only way that you could exercise your long put to sell the stock AFTER the short call has expired would be if the put expired at a later date than the short call.