For Question 1. It is documented in Corolary 4.2.4.8 in Lurie's Higher Topos theory that $N_\Delta(\underline{M}^{cf})$ is an $\infty$-category with small limits (and small colimits). Moreover, the inclusion $M^{cf}\subset \underline{M}^{cf}$ induces a functor $$(*)\qquad L(M,W)\cong L(M^{cf},W\cap M^{cf})\to N_\Delta(\underline{M}^{cf})\, .$$ The homotopy $1$-category of the $\infty$-category $N_\Delta(\underline{M}^{cf})$ is the ordinary $1$-localization of $M^{cf}$ by weak equivalences. This means that the comparison functor above induces an equivalence on homotopy categories. On the other hand, since, by virtue of Proposition 7.7.4 of Higher categories and homotopical algebra, $L(M,W)$ also has small (co)limits, it is sufficient to prove that the functor $(*)$ above commutes with finite limits, by Theorem 6.7.10 from loc. cit. The fact that the terminal object is preserved is obvious. Using Proposition 7.5.6 (as well as Theorem 7.5.18) from loc. cit., this amounts to check that pullback squares along fibrations between fibrant objects do provide pulback squares in $N_\Delta(\underline{M}^{cf})$, which follows from Theorem 4.2.4.1 in Higher Topos theory.

In fact, with the arguments above, one may characterize $L(M,W)$ as follows: an $\infty$-category $M_\infty$ equipped with a functor $\gamma: N(M)\to M_\infty$ is the localization of $M$ by $W$ if and only if the following conditions are verified:

1. $\gamma$ sends any element of $W$ to an invertible map in $M_\infty$;
2. the induced functor $M\to ho(M_\infty)$ is the $1$-localization of $M$ by weak equivalences;
3. the functor preserves terminal objects and sends any pullback square of fibrant objects along fibrations in $M$ to pullback squares in $M_\infty$.

(the latter characterization can be extended to all kind of variations on model structures that I know of: semi-model structures, categories of fibrant objects, and so on).

For Question 2. I insist that checking that the hammock localization of Dwyer and Kan has the universal property of $L(C,W)$ really is obvious from the original construction via the Quillen equivalence relating relative categories, simplicial categories, and quasi-categories, just by reading the original sources. But there are documented more recent references such as Hinich's Dwyer-Kan localization revisited, for instance.

For Question 1. It is documented in Corolary 4.2.4.8 in Lurie's Higher Topos theory that $N_\Delta(\underline{M}^{cf})$ is an $\infty$-category with small limits (and small colimits). Moreover, the inclusion $M^{cf}\subset \underline{M}^{cf}$ induces a functor $$(*)\qquad L(M,W)\cong L(M^{cf},W\cap M^{cf})\to N_\Delta(\underline{M}^{cf})\, .$$ The homotopy $1$-category of the $\infty$-category $N_\Delta(\underline{M}^{cf})$ is the ordinary $1$-localization of $M^{cf}$ by weak equivalences. This means that the comparison functor above induces an equivalence on homotopy categories. On the other hand, since, by virtue of Proposition 7.7.4 of Higher categories and homotopical algebra, $L(M,W)$ also has small (co)limits, it is sufficient to prove that the functor $(*)$ above commutes with finite limits, by Theorem 6.7.10 from loc. cit. The fact that the terminal object is preserved is obvious. Using Proposition 7.5.6 (as well as Theorem 7.5.18) from loc. cit., this amounts to check that pullback squares along fibrations between fibrant objects do provide pulback squares in $N_\Delta(\underline{M}^{cf})$, which follows from Theorem 4.2.4.1 in Higher Topos theory.

In fact, with the arguments above, one may characterize $L(M,W)$ as follows: a finitely complete $\infty$-category $M_\infty$ equipped with a functor $\gamma: N(M)\to M_\infty$ is the localization of $M$ by $W$ if and only if the following conditions are verified:

1. $\gamma$ sends any element of $W$ to an invertible map in $M_\infty$;
2. the induced functor $M\to ho(M_\infty)$ is the $1$-localization of $M$ by weak equivalences;
3. the functor preserves terminal objects and sends any pullback square of fibrant objects along fibrations in $M$ to pullback squares in $M_\infty$.

(the latter characterization can be extended to all kind of variations on model structures that I know of: semi-model structures, categories of fibrant objects, and so on).

For Question 2. I insist that checking that the hammock localization of Dwyer and Kan has the universal property of $L(C,W)$ really is obvious from the original construction via the Quillen equivalence relating relative categories, simplicial categories, and quasi-categories, just by reading the original sources. But there are documented more recent references such as Hinich's Dwyer-Kan localization revisited, for instance.

For Question 1. It is documented in Corolary 4.2.4.8 in Lurie's Higher Topos theory that $N_\Delta(\underline{M}^{cf})$ is an $\infty$-category with small limits (and small colimits). Moreover, the inclusion $M^{cf}\subset \underline{M}^{cf}$ induces a functor $$(*)\qquad L(M,W)\cong L(M^{cf},W\cap M^{cf})\to N_\Delta(\underline{M}^{cf})\, .$$ The homotopy $1$-category of the $\infty$-category is the ordinary $1$-localization of $M^{cf}$ by weak equivalences. This means that the comparison functor above induces an equivalence on homotopy categories. On the other hand, since, by virtue of Proposition 7.7.4 of Higher categories and homotopical algebra, $L(M,W)$ also has small (co)limits, it is sufficient to prove that the functor $(*)$ above commutes with finite limits, by Theorem 6.7.10 from loc. cit. The fact that the terminal object is preserved is obvious. Using Proposition 7.5.6 (as well as Theorem 7.5.18) from loc. cit., this amounts to check that pullback squares along fibrations between fibrant objects do provide pulback squares in $N_\Delta(\underline{M}^{cf})$, which follows from Theorem 4.2.4.1 in Higher Topos theory.

In fact, with the arguments above, one may characterize $L(M,W)$ as follows: an $\infty$-category $M_\infty$ equipped with a functor $\gamma: N(M)\to M_\infty$ is the localization of $M$ by $W$ if and only if the following conditions are verified:

1. $\gamma$ sends any element of $W$ to an invertible map in $M_\infty$;
2. the induced functor $M\to ho(M_\infty)$ is the $1$-localization of $M$ by weak equivalences;
3. the functor preserves terminal objects and sends any pullback square of fibrant objects along fibrations in $M$ to pullback squares in $M_\infty$.

(the latter characterization can be extended to all kind of variations on model structures that I know of: semi-model structures, categories of fibrant objects, and so on).

For Question 2. I insist that checking that the hammock localization of Dwyer and Kan has the universal property of $L(C,W)$ really is obvious from the original construction via the Quillen equivalence relating relative categories, simplicial categories, and quasi-categories, just by reading the original sources. But there are documented more recent references such as Hinich's Dwyer-Kan localization revisited, for instance.

For Question 1. It is documented in Corolary 4.2.4.8 in Lurie's Higher Topos theory that $N_\Delta(\underline{M}^{cf})$ is an $\infty$-category with small limits (and small colimits). Moreover, the inclusion $M^{cf}\subset \underline{M}^{cf}$ induces a functor $$(*)\qquad L(M,W)\cong L(M^{cf},W\cap M^{cf})\to N_\Delta(\underline{M}^{cf})\, .$$ The homotopy $1$-category of the $\infty$-category $N_\Delta(\underline{M}^{cf})$ is the ordinary $1$-localization of $M^{cf}$ by weak equivalences. This means that the comparison functor above induces an equivalence on homotopy categories. On the other hand, since, by virtue of Proposition 7.7.4 of Higher categories and homotopical algebra, $L(M,W)$ also has small (co)limits, it is sufficient to prove that the functor $(*)$ above commutes with finite limits, by Theorem 6.7.10 from loc. cit. The fact that the terminal object is preserved is obvious. Using Proposition 7.5.6 (as well as Theorem 7.5.18) from loc. cit., this amounts to check that pullback squares along fibrations between fibrant objects do provide pulback squares in $N_\Delta(\underline{M}^{cf})$, which follows from Theorem 4.2.4.1 in Higher Topos theory.

In fact, with the arguments above, one may characterize $L(M,W)$ as follows: an $\infty$-category $M_\infty$ equipped with a functor $\gamma: N(M)\to M_\infty$ is the localization of $M$ by $W$ if and only if the following conditions are verified:

1. $\gamma$ sends any element of $W$ to an invertible map in $M_\infty$;
2. the induced functor $M\to ho(M_\infty)$ is the $1$-localization of $M$ by weak equivalences;
3. the functor preserves terminal objects and sends any pullback square of fibrant objects along fibrations in $M$ to pullback squares in $M_\infty$.

(the latter characterization can be extended to all kind of variations on model structures that I know of: semi-model structures, categories of fibrant objects, and so on).

For Question 2. I insist that checking that the hammock localization of Dwyer and Kan has the universal property of $L(C,W)$ really is obvious from the original construction via the Quillen equivalence relating relative categories, simplicial categories, and quasi-categories, just by reading the original sources. But there are documented more recent references such as Hinich's Dwyer-Kan localization revisited, for instance.