Euler path and Euler circuit
No, a graph cannot contain Euler path and Euler circuit both.
Because for Euler path the graph must have exactly two vertices having odd degree but for Euler circuit all the vertices in that graph must have even degree.
Hamiltonian Path and Hamiltonian circuit
Yes, a graph can have both hamiltonian path and circuit.
Imagine a triangle ABC.
Then the path ABC is a hamiltonian path and the circuit ABCA is a hamiltonian circuit.