Finding a string with some specified hamming distances from other strings
Your problem is likely hard. I will consider a decision version of your problem in which the goal is to ask whether there exists any solution. I will show that the binary variant is NP-complete, and the same ideas probably imply that this decision version of your problem is also NP-complete.

Other decision versions of your problem are likely also hard, though perhaps not NP-hard. For example, consider the problem in which you are promised that a unique solution exists, and your goal is to determine the value of its first bit. This problem cannot be NP-hard since it has the same complexity as its complement. Nevertheless, it is probably hard in some other, appropriate sense.

The decision version of your problem that I will analyze asks whether there is an answer.
The problem is clearly in NP, and we prove that it is NP-hard by reduction from X3SAT, a variant of 3SAT in which a formula is satisfiable if every clause contains exactly one satisfied literal. Given a 3CNF on $n$ variables $x_1,\ldots,x_n$, we will consider strings of length $2n$ which encode a truth assignment: a 1 in position $2(i-1)+1$ encodes that $x_i$ is true, and a 1 in position $2(i-1)+2$ encodes that $x_i$ is false.

In order to force that any solution must have exactly one 1 in each of the positions $2(i-1)+1,2(i-1)+2$, we ask that it have distance $n$ from the strings $(00)^n$ and $(00)^{i-1}(11)(00)^{n-i}$ for $1 \leq i \leq n$. In order to force any solution to satisfy all clauses, for every clause we take the string which has 1s in the positions corresponding to the literals in the clause, and we ask that the solution be at distance $(n-3)+2$ from this string. A string satisfies these $n+m+1$ constraints (where $m$ is the number of clauses) if and only if it is a satisfying assignment to the given 3CNF.