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As shown by Gerstenhaber and Voronov [Higher operations on the Hochschild complex], the Hochschild complex of an associative algebra is endowed with a natural structure of brace algebra. The first brace operation identifies with the Gerstenhaber product while the second encodes the obstruction for the Gerstenhaber product to be associative. It is also used to define the cup product.

1)The Gerstenhaber bracket possesses a natural interpretation using the bar construction as a commutator of coderivations of the tensor algebra associated with the associative algebra. Is there a similar bar construction interpretation of the brace operations ?

The cohomology of the brace algebra is endowed with a natural structure of a Gerstenhaber algebra.

2)Is there a $n$-version of the concept of brace algebra such that its cohomology is a $n$-Gerstenhaber algebra ?

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Yes. A good reference is Damien Calacque and Thomas Willwacher's Triviality of the higher Formality Theorem

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