What is the ruling on buying and gifting religious tapes to a neighbor whose husband is traveling, knowing that she appears before me and speaks with me about matters of religion?

There is no harm in gifting beneficial Islamic tapes to a non-mahram woman, provided that they are delivered through one of the giver's female mahrams, in order to avoid suspicion and temptation. Furthermore, it is not permissible for a woman to appear before a man without full hijab, nor is it permissible to be in seclusion with her.