Does combinational and sequential logic correspond to some mathematical logic systems?
Is it correct that the functionalities of digit circuits are divided into combinational logic and sequential logic?

Yes. And sequential logic in digital electronics is subdivided into asynchronous and synchronous sequential logic.

Is combinational logic the same thing as propositional logic in mathematical logic?

No, they are not the same. Propositional logic operates on propositions, which are declarative sentences. In digital electronics, combinational logic refers to circuits which implement Boolean operations, and whose outputs are a function of the current inputs. Hence it is time-independent. Boolean operations can be operated only on Boolean data types. Boolean data types can take only 2 values.

Are finite state machine/automaton and temporal/modal/... logic different (unrelated) approaches to study sequential logic?

Not entirely. Sequential Logic is built from combinational logic circuits and has memory. In sequential logic, the outputs will also depend on the previous inputs. Sequential circuits have feedback.
Finite State Machines are sequential logic since they have memory. An automaton can be implemented in sequential logic if it has a finite number of states and is deterministic.
There are also automata which are undecidable, and it is not possible to build sequential logic for these automata. For example, Conway's game of life.
Strictly speaking, only deterministic Finite State machines can be built using sequential logic.
Non-determinism can be emulated using Pseudorandom Number Generation. This is not true non-determinism.

Do combinational and sequential logics correspond to some mathematical logic systems?

Combinational logic directly corresponds to Boolean Algebra, which is an algebraic logic.
Sequential logic does not correspond to any mathematical logic system. Sequential logic does correspond to the strictly non-deterministic finite state machine in automata theory, which is considered to be a field of computer science.