Is there a minimum limit for No. of photons that Alice would send to Bob in a QKD scheme like BB84? If not than what would be a desirable length for initial qubit string?
The minimum limit would be 1, which transmits a key of length 1 half of the time. Ideally, you would use hardware capable of conveying this information with a single photon per bit in the string, so one photon.
The string length is a practical consideration, not a cryptographic one. Sending fewer bits means you will have to re-key more often. Sending 1 bit at a time would basically be equivalent to creating a stream of key information. Size the qubit string to fit your application's practical needs.