Is it compulsory for a person studying the Vedas to remove all facial hair and just have a Shikha/shendi?
It is recommended not to have facial hairs at least while one is reciting the Vedas. We have the following reference from the Dharma Sutras of Baudhayana 4.5.1-3:
Now I will proclaim the rites of the Rg-, Yajur-, and Atharva-vedas by which a man will quickly obtain his heart’s desires.
2 He should perform the rites only after purifying his body by the soft recitation of prayers, burnt offerings, oblations, and restraints. He will not attain success otherwise.
When he is about to engage in the soft recitation of prayers, make burnt offerings and oblations, and practise restraints, a Brahmin should first get his head and beard shaved and his nails clipped on an auspicious day and under an auspicious lunar mansion during the fortnight