It happens immediately at the opening trade when it gets filled!...

Ie. with an opening ShortPut fill, the unsettledCash immediately rises to the value "(K + Premium) * 100"

You tell me the logic in this, and whether the above formula used by TDA is correct...
Actually they subtract the premium + fees from that value, ie. $0.65 + $0.01 per contract, for example, ie.:
K=3.50, Premium=0.40 --> gives unsettledCash = (3.50 + 0.40) * 100 - 0.65 - 0.01 = $389.34
Beware: as stated, this is a CashSecuredPut (CSP), ie. a ShortPut with an implicit CashCollateral = K * 100
Reminder: we have 2 issues here: the unsettledCash case and the tax case, yes.