When translating modal logic to first-order logic, I notice that the translated structure in first-order logic does not have constant symbols. I am thinking about whether it is possible that the translated structure in first-order logic has some constant symbols.
Because the constant symbols can be equivalently replaced with a unary predicate, if the translated structure has a constant $c$, that constant can be replaced with a unary predicate $P(x,c)=:(x=c)$. Then the corresponding modal logic has an extra proposition $c$ and valuation $V(w) = \{ c \}$.
I want to know if what I think for constant symbols is correct or not. And if there is a function in a translated structure, does there exist a corresponding modal logic?
I think that replacing constants with predicates is a bit more involved than you present. See, for example, here is first-order logic with constants equally expressive as first-order logic without constants?
Regarding the standard translation, your approach with equality is actually a part of standard translation of hybrid logic into a fragment of FOL with equality (without functions and constants).
There are some translations that have functions in their image. See, for example, first-order intensional modal logic http://cachescan.bcub.ro/e-book/V/580527_07.pdf