What is the duration of a daily Pradosham? What are the Dos and Don’ts during that period? Can we do Veda Parayanam at temples during such periods and anadhyayana period? Are there special rules for Maha-Pradosham? Can we do parayanam on that day at the temple?

The daily sunset time is the pradosham period. At that time sandhyavandanam has to be performed; one should not learn Veda, sleep, etc., at that time.
Veda parayanam can be done if it is an ongoing recitation at the temple.
Special rules are applicable for Maha Pradosham. One should not recite Vishnu Sahasranama; it can be recited during a temple utsavam or recitations organized at the temple. Observing silence is prescribed for Maha-Pradosham, so if there is a recitation at the temple one can recite so.
During anadhyayana period, Veda Parayanam and other parayanam can be done at the temples. One should not recite them at home.
But there is nothing specifically called daily pradosham.