How do Othello's themes of religion, culture, racism and gender inequality affect the title character's relationship with the Venetians?

Good question!

1. Religion--while Othello is not as overtly religious as The Merchant of Venice. The play relies heavily on Garden of Eden allusions.  The original battle between good and evil repeats itself within the play.  Desdemona (Eve) and Othello (Adam) are tempted by the wily serpent (Iago) who eventually causes their "fall."  In regards to this idea of good and evil relating to Othello's relationship with the Venetians, this theme is almost as important as Shakespeare's discussion of race.  The Venetians, though quite forward-thinking folks for their time period--tend to view Othello as evil (or "black at heart").  When the play opens, Brabantio and Iago play on this prejudice of the Venetians by referencing Othello's supposed magical or satanic powers.  Because the Venetians in the play are blind to true evil (Iago), they readily misjudge Othello.

2. Culture--during the play's time setting, Venice was the cosmopolitan center of trade and culture.  Because of this, the Venetians could be more accepting of other cultures and races in the sense that they allowed foreigners relative freedom.  However, those foreigners--such as Othello--were never truly accepted.  While the Duke and Senators dine with Othello, seek his advice, and invite him to various social events, they do not embrace him, show an interest in his culture, or want him marrying their daughters.  They use him for his military skills and would not allow him to play a role in their lives or culture at all if not for those skills.

3. Racism, of course, affects Othello's relationship with almost everyone, including his wife.  Desdemona is certainly not racist, but because of Othello's race and the prejudice that he must constantly fight against, he doubts his wife's love and loyalty.  He finds it easy to believe false charges against Desdemona because he does not feel worthy of her.  This feeling is the result of the racist treatment he has faced throughout his career in Venice. Likewise, Iago has no difficulty portraying Othello as a lust-filled older man taking advantage of a young girl's innocence.  He convinces Brabantio that Othello must have used magical powers to entice Desdemona into marriage, for why else would a noble white girl marry a black military leader?  This is the prejudice of Iago's time; so no one argues with his words.

4. Gender inequality--this theme does not affect Othello's relationship with the Venetians as much as others, but it does play a part in his marriage. Even though he and Desdemona truly love one another, Othello resorts to the domineering husband role of the time period and silences Desdemona.  Perhaps in a more modern setting, she would have been more persuasive as an equal partner in the relationship.  This inequality also plays a part in Iago and Emilia's relationship.  Even though Iago humiliates Emilia in public, she still feels an obligation to her husband to obey his "orders" to obtain Desdemona's handkerchief.  Obviously, Emilia is loyal to her mistress, but her loyalty as a dominated wife overrules the first relationship.  She is intelligent and must know that Iago has no good intention for his use of the handkerchief, yet she follows through with giving him the item, which eventually leads to the play's horrific final events.