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Yes, by compactness.

Let $R$ denote your favorite hyperreal ordered field and let $\delta\in R$ be a positive infinitesimal. Let $\mathcal{E}$ denote the set of all (standard) finite Boolean subalgebras of $\mathcal{P}([0,1))$. For every $A\in\mathcal{E}$, let $\lambda_A(I)$ be the (exact) length of $I$ for all half-open intervals $I\in A$; for all open or closed intervals $I\in A$, respectively subtract or add $\delta$ to the length of $I$ to define $\lambda_A(I)$; let $\lambda_A(S)=\delta$ for all singletons $S\in A$. Extend $\lambda_A$ to a probability measure $\mu_A$ on $A$.

(Specifically, for each minimal interval $I\in A$ that is not an atom of $A$, partition $I$ into its atomic subsets $H_0,\ldots,H_n$ such the singletons among $H_0,\ldots,H_n$ are $H_0,\ldots H_{m-1}$. Choose a positive $\mu_A(H_i)\in R$ for each $i\geq m$, such that $\sum_{i=m}^n\mu_A(H_i)=\lambda_A(I)-m\delta$. For all atomic $K\in A\cap\mathrm{dom}(\lambda_A)$, define $\mu_A(K)=\lambda_A(K)$. Now extend $\mu_A$ from the atoms to all of $A$.)

Let $U$ be a fine ultrafilter on $\mathcal{E}$. ("Fine" means that $\{B\in\mathcal{E}:A\subseteq B\}\in U$ for all $A\in\mathcal{E}$.) The ultraproduct measure $\mu_U$ is $R^U$-valued and has the two properties you seek.

Yes, by compactness.

Let $R$ denote your favorite hyperreal ordered field and let $\delta\in R$ be a positive infinitesimal. Let $\mathcal{E}$ denote the set of all (standard) finite Boolean subalgebras of $\mathcal{P}([0,1))$. For every $A\in\mathcal{E}$, let $\lambda_A(I)$ be the (exact) length of $I$ for all half-open intervals $I\in A$; for all open or closed intervals $I\in A$, respectively subtract or add $\delta$ to the length of $I$ to define $\lambda_A(I)$; let $\lambda_A(S)=\delta$ for all singletons $S\in A$. Extend $\lambda_A$ to a probability measure $\mu_A$ on $A$.

(Specifically, for each minimal finite union of intervals $F\in A$, let the connected components of $F$ be $[a_0,b_0),\ldots,[a_k,b_k)$ with $p$ elements of $\{a_i,b_i:i\leq k\}$ added and $q$ removed. Partition $F$ into its atomic subsets $H_0,\ldots,H_n$. Choose a positive $\mu_A(H_i)\in R$ for each $i$, such that $\sum_{i\leq n}\mu_A(H_i)=(p-q)\delta+\sum_{i\leq k}(b_i-a_i)$. Now extend $\mu_A$ from the atoms to all of $A$.)

Let $U$ be a fine ultrafilter on $\mathcal{E}$. ("Fine" means that $\{B\in\mathcal{E}:A\subseteq B\}\in U$ for all $A\in\mathcal{E}$.) The ultraproduct measure $\mu_U$ is $R^U$-valued and has the two properties you seek.

Yes, by compactness.

Let $R$ denote your favorite hyperreal ordered field and let $\delta\in R$ be a positive infinitesimal. Let $\mathcal{E}$ denote the set of all (standard) finite Boolean subalgebras of $\mathcal{P}([0,1))$. For every $A\in\mathcal{E}$, let $\lambda_A(I)$ be the (exact) length of $I$ for all half-open intervals $I\in A$; for all open or closed intervals $I\in A$, respectively subtract or add $\delta$ to the length of $I$ to define $\lambda_A(I)$; let $\lambda_A(S)=\delta$ for all singletons $S\in A$. Extend $\lambda_A$ to a probability measure $\mu_A$ on $A$.

(Specifically, for each minimal interval $I\in A$ that is not an atom of $A$, partition $I$ into its atomic subsets $H_0,\ldots,H_n$ such the singletons among $H_0,\ldots,H_n$ are $H_0,\ldots H_{m-1}$. Choose a positive $\mu_A(H_i)\in R_1$ for each $i\geq m$, such that $\sum_{i=m}^n\mu_A(H_i)=\lambda_A(I)-m\delta$. For all atomic $K\in A\cap\mathrm{dom}(\lambda_A)$, define $\mu_A(K)=\lambda_A(K)$. Now extend $\mu_A$ from the atoms to all of $A$.)

Let $U$ be a fine ultrafilter on $\mathcal{E}$. ("Fine" means that $\{B\in\mathcal{E}:A\subseteq B\}\in U$ for all $A\in\mathcal{E}$). The ultraproduct measure $\mu_U$ is $R^U$-valued and has the two properties you seek.

Yes, by compactness.

Let $R$ denote your favorite hyperreal ordered field and let $\delta\in R$ be a positive infinitesimal. Let $\mathcal{E}$ denote the set of all (standard) finite Boolean subalgebras of $\mathcal{P}([0,1))$. For every $A\in\mathcal{E}$, let $\lambda_A(I)$ be the (exact) length of $I$ for all half-open intervals $I\in A$; for all open or closed intervals $I\in A$, respectively subtract or add $\delta$ to the length of $I$ to define $\lambda_A(I)$; let $\lambda_A(S)=\delta$ for all singletons $S\in A$. Extend $\lambda_A$ to a probability measure $\mu_A$ on $A$.

(Specifically, for each minimal interval $I\in A$ that is not an atom of $A$, partition $I$ into its atomic subsets $H_0,\ldots,H_n$ such the singletons among $H_0,\ldots,H_n$ are $H_0,\ldots H_{m-1}$. Choose a positive $\mu_A(H_i)\in R$ for each $i\geq m$, such that $\sum_{i=m}^n\mu_A(H_i)=\lambda_A(I)-m\delta$. For all atomic $K\in A\cap\mathrm{dom}(\lambda_A)$, define $\mu_A(K)=\lambda_A(K)$. Now extend $\mu_A$ from the atoms to all of $A$.)

Let $U$ be a fine ultrafilter on $\mathcal{E}$. ("Fine" means that $\{B\in\mathcal{E}:A\subseteq B\}\in U$ for all $A\in\mathcal{E}$.) The ultraproduct measure $\mu_U$ is $R^U$-valued and has the two properties you seek.

Yes, by compactness.

Let $R$ denote your favorite hyperreal ordered field and let $\delta\in R$ be a positive infinitesimal. Let $\mathcal{E}$ denote the set of all (standard) finite Boolean subalgebras of $\mathcal{P}([0,1))$. For every $A\in\mathcal{E}$, let $\lambda_A(I)$ be the (exact) length of $I$ for all half-open intervals $I\in A$; let $\lambda_A(S)=\delta$ for all singletons $S\in A$. Extend $\lambda_A$ to a probability measure $\mu_A$ on $A$.

(Specifically, for each minimal half-open interval $I\in A$ that is not an atom of $A$, partition $I$ into its atomic subsets $H_0,\ldots,H_n$ such the singletons among $H_0,\ldots,H_n$ are $H_0,\ldots H_{m-1}$. Choose a positive $\mu_A(H_i)\in R_1$ for each $i\geq m$, such that $\sum_{i=m}^n\mu_A(H_i)=\lambda_A(I)-m\delta$. For all atomic $K\in A\cap\mathrm{dom}(\lambda_A)$, define $\mu_A(K)=\lambda_A(K)$. Now extend $\mu_A$ from the atoms to all of $A$.)

Let $U$ be a fine ultrafilter on $\mathcal{E}$. ("Fine" means that $\{B\in\mathcal{E}:A\subseteq B\}\in U$ for all $A\in\mathcal{E}$). The ultraproduct measure $\mu_U$ is $R^U$-valued and has the two properties you seek.

Yes, by compactness.

Let $R$ denote your favorite hyperreal ordered field and let $\delta\in R$ be a positive infinitesimal. Let $\mathcal{E}$ denote the set of all (standard) finite Boolean subalgebras of $\mathcal{P}([0,1))$. For every $A\in\mathcal{E}$, let $\lambda_A(I)$ be the (exact) length of $I$ for all half-open intervals $I\in A$; for all open or closed intervals $I\in A$, respectively subtract or add $\delta$ to the length of $I$ to define $\lambda_A(I)$; let $\lambda_A(S)=\delta$ for all singletons $S\in A$. Extend $\lambda_A$ to a probability measure $\mu_A$ on $A$.

(Specifically, for each minimal interval $I\in A$ that is not an atom of $A$, partition $I$ into its atomic subsets $H_0,\ldots,H_n$ such the singletons among $H_0,\ldots,H_n$ are $H_0,\ldots H_{m-1}$. Choose a positive $\mu_A(H_i)\in R_1$ for each $i\geq m$, such that $\sum_{i=m}^n\mu_A(H_i)=\lambda_A(I)-m\delta$. For all atomic $K\in A\cap\mathrm{dom}(\lambda_A)$, define $\mu_A(K)=\lambda_A(K)$. Now extend $\mu_A$ from the atoms to all of $A$.)

Let $U$ be a fine ultrafilter on $\mathcal{E}$. ("Fine" means that $\{B\in\mathcal{E}:A\subseteq B\}\in U$ for all $A\in\mathcal{E}$). The ultraproduct measure $\mu_U$ is $R^U$-valued and has the two properties you seek.