In his 2001 paper Sets, Classes and Categories, F. A. Muller lays out a modification of Ackermann class theory that he claims is not conservative over Ackermann's original theory in the sense that it proves the consistency of $ZFC$ (he adds choice to his modification in addition to other changes), while Ackermann's theory $A$ was equiconsistent with $ZF$ ($ZFC$ if we add choice to $A$). A brief description of the axioms of Muller's theory can be found here.

1. Is this claim correct?

He cites another work of his, 'Structures for Everyone. Contemplations and Proofs in the Foundations and Philosophy of Physics and Mathematics', as the source for a proof but I don't have access to the text for proof verification. In the comments section of an answer to an older MO question about Ackermann's original theory the answerer contradicts Muller and claims that his theory is conservative over $ZFC$; hopefully one of the experts here can put this issue to rest for good.

If his modified theory is a step up in consistency strength from $ZFC$,

2. Does this remain true if we add large cardinal axioms to both theories? That is, letting $SC$ denote the theory in Muller's paper and $\mathbb{V}$ denote the universe of sets in this theory, for a large cardinal axiom $\phi$ do we have that $SC+\phi^\mathbb{V}$ is a step up in consistency strength from $ZFC+\phi$?

In his 2001 paper Sets, Classes and Categories, F. A. Muller lays out a modification of Ackermann class theory that he claims is not conservative over Ackermann's original theory in the sense that it proves the consistency of $ZFC$ (he adds choice to his modification in addition to other changes), while Ackermann's theory $A$ was equiconsistent with $ZF$ ($ZFC$ if we add choice to $A$). A brief description of the axioms of Muller's theory can be found here.

1. Is this claim correct?

He cites another work of his, 'Structures for Everyone. Contemplations and Proofs in the Foundations and Philosophy of Physics and Mathematics', as the source for a proof but I don't have access to the text for proof verification. In the comments section of an answer to an older MO question about Ackermann's original theory the answerer contradicts Muller and claims that his theory is conservative over $ZFC$; hopefully one of the experts here can put this issue to rest for good.

If his modified theory is a step up in consistency strength from $ZFC$,

2. Does this remain true if we add large cardinal axioms to both theories? That is, letting $SC$ denote the theory in Muller's paper and $\mathbb{V}$ denote the universe of sets in this theory, for a large cardinal axiom $\phi$ do we have that $SC+\phi^\mathbb{V}$ is a step up in consistency strength from $ZFC+\phi$?