There seems to be a mistake in the construction from "An enrichment..." See http://arxiv.org/pdf/1104.3441v1 page 3. That paper gives an alternative construction of a symmetric spectrum representing (equivariant) K-theory.

Concerning question 2: the induced functor $\mathrm{KK}\to\mathrm{Der}(\mathbf{K})$ is fully faithful on the bootstrap class of Rosenberg--Schochet (the localizing subcategory generated by $\mathbb C$). It cannot be fully faithful on all of $\mathrm{KK}$ because there are counterexamples to the Universal Coefficient Theorem in $\mathrm{KK}$ but not in $\mathrm{Der}(\mathbf{K})$.

There seems to be a mistake in the construction from "An enrichment..." See http://arxiv.org/pdf/1104.3441v1 page 3. That paper gives an alternative construction of a symmetric spectrum representing (equivariant) K-theory.

Concerning question 2: the induced functor $\mathrm{KK}\to\mathrm{Der}(\mathbf{K})$ is fully faithful and strongly monoidal on the bootstrap class of Rosenberg--Schochet (the localizing subcategory generated by $\mathbb C$). It cannot be fully faithful on all of $\mathrm{KK}$ because there are counterexamples to the Universal Coefficient Theorem in $\mathrm{KK}$ but not in $\mathrm{Der}(\mathbf{K})$.

There seems to be a mistake in the construction from "An enrichment..." See http://arxiv.org/pdf/1104.3441v1 page 3. That paper gives an alternative construction for symmetric spectra.

Concerning question 2: the induced functor $\mathrm{KK}\to\mathrm{Der}(\mathbf{K})$ is fully faithful on the bootstrap class of Rosenberg--Schochet (the localizing subcategory generated by $\mathbb C$). It cannot be fully faithful on all of $\mathrm{KK}$ because there are counterexamples to the Universal Coefficient Theorem in $\mathrm{KK}$ but not in $\mathrm{Der}(\mathbf{K})$.

There seems to be a mistake in the construction from "An enrichment..." See http://arxiv.org/pdf/1104.3441v1 page 3. That paper gives an alternative construction of a symmetric spectrum representing (equivariant) K-theory.

Concerning question 2: the induced functor $\mathrm{KK}\to\mathrm{Der}(\mathbf{K})$ is fully faithful on the bootstrap class of Rosenberg--Schochet (the localizing subcategory generated by $\mathbb C$). It cannot be fully faithful on all of $\mathrm{KK}$ because there are counterexamples to the Universal Coefficient Theorem in $\mathrm{KK}$ but not in $\mathrm{Der}(\mathbf{K})$.