When Dr Kareem's two henchmen are in a jeep looking for Jonny and Hadji, one of them mistakenly refers to Hadji as "the Hindu boy." Hadji is clearly a Sikh, not a Hindu, because he wears a Sikh turban. Most fans also know his full name is Hadji Singh, Singh being a Sikh name.
A mummy's wrappings would be discolored and stiffened by a combination of age and the mummification process. The mummy's bandages are white and as flexible as new cloth. Furthermore, the mummy itself would not be held to be the actual god Anubis; while a mummified individual may have been deified after death, they were not part of the standard Egyptian pantheon.
When the thieves show Dr. Kareem the statue, their instructions are to take it "...across the border to the ruins of Giza". However, the establishing shot shows that the temple of Anubis is near both the Great Sphinx and the pyramid of Khafre, which is in Giza. Therefore, they're taking the statue back where it came from, and no border crossings are involved. If the borders of the various governorates of Egypt are the ones under consideration, they would not be a barrier to travel.
Dogs are considered ritually unclean in Islamic culture. Taking Bandit on a trip to Egypt was not a smart move.