Is it permissible for a sister reciting Qur'aan at home without Khimaar and sock? If so would she get the same blessing when wearing the Khimaar and sock?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

There is no harm for a woman to recite the Qur'an without wearing a veil and socks if she is not in the presence of a marriageable man. For more details, please refer to Fatwa: 85384.

As regards whether she has the same reward as when she wears the veil, if she wears the veil when reading the Qur'an as a sign of glorification and honour for the Words of Allaah, we hope that she will get more reward because of the honour that she has for Allaah's Book. Moreover, some scholars stated that among the etiquettes of reading the Qur'an is that a person should wear his best attire. 

Allaah knows best.