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This assertion is always true. Synthetically, either bracketing places three disjoint blocks of order types $\alpha ,\beta ,\gamma$ in that order. The identity on these blocks is an order isomorphism, so both sums have the same order type. This proves associativity of ordinal addition for successor and limit ordinals alike.

True. This is left distributivity of ordinal multiplication. Fix $\alpha ,\beta$ and induct on $\gamma$. The zero case is immediate. The successor step uses $\alpha (\eta +1)=\alpha \eta +\alpha$ and associativity of ordinal addition. At a limit, multiplication and addition are continuous in their right argument, so taking suprema of the earlier equalities gives $\alpha (\beta +\gamma )=\alpha \beta +\alpha \gamma$.

True. Every $\beta <{\omega }_1$ is a countable ordinal, andWriting any $\alpha \ge {\omega }_1$ as ${\omega }_1+\delta$ and using associativity gives