Does a path can be Hamiltonian and Eulerian at the same time? If so does it force it to be a simple circle? 
Or any other restrictions?
How would it look like?
Thanks in advance
 A: A path is Hamiltonian if each vertex is visited exactly once.  A path is Eulerian if every edge is traversed exactly once.  Clearly, these conditions are not mutually exclusive for all graphs:  if a simple connected graph $G$ itself consists of a path (so exactly two vertices have degree $1$ and all other vertices have degree $2$), then that path is both Hamiltonian and Eulerian.  If $G$ is a cycle, then that cycle is Hamiltonian and Eulerian.
A: The answer is yes.
We have two criteria to meet:

$1.$ Have either $2$ odd vertices or have none at all.
$2.$ Travel to each vertex once and only once, and return to the starting point.

These can both be met, and an example is:

(Source: "Eulerian and Hamiltonian Graphs", Mathematics Learning Center, p.3)
Note: I have assumed for criteria $2$ that you are referring to a Hamiltonian circuit.
A: To set the record clear: Yes.
A Path can be both Eularian and Hamiltonian. A Hamiltonian path is a spanning path, and an Eularian path goes through each edge exactly once. To consider a path holding both properties at the same time, think of the maximal path in $P_n$.
