a) Will a polyhedron be formed if its faces are irregular polygons? If so, will it be called an irregular polyhedron?
b) Will a polyhedron be formed if two separate polyhedrons are joined with each other by a face, though all of the faces are polygons itself?
ad a)
Regularity of a polytope in N dimensions (or of a polyhedron within 3 dimensions) is defined to mean that each flag (consisting of a vertex, an incident edge, a face incident to the latter, etc.) could be mapped by symmetry onto any other flag. Therefore the regularity of a polyhedron always implies the regularity of its face polygons. Or, reversing this statement, the irregularity of any single face already implies the irregularity of the polyhedron.
ad b)
You surely can adjoin 2 polyhedra at a (then to be reduced) common face. E.g. stack 2 cubes atop each other. Whether such an outcome generally still follows your chosen definition of "polyhedra" depends of the thereby implied restrictions. For instance if you attach 2 dodecahedra atop each other, then the outcome no longer will be convex. Thus, if your chosen definition of polyhedra silently assumes convexity, then that latter example won't be one.
--- rk