Are these things really proof of anything?

No, the handkerchief and Iago's incessant innuendos are proof of nothing. Othello hangs off Iago's words because he thinks in absolutes. Either Desdemona is a pure angel or she is a whore: there is no in-between. Othello's simple psyche makes fertile ground for Iago to plant his lies and water them with circumstantial evidence.