Can bending spacetime produce electromagnetic fields? Let's say that we can somehow bent spacetime, can this produce electromagnetic fields? or EMF can only be emitted from charged particles of matter?
 A: Only from charges. It's the quantum property that couples to photons. 
Gravity couples to anything, but there has to be some charge acccelerated to create photons. So gravity could accelerate it, but it also would depend on what is your frame of reference as to whether you'd see it. 
Another way to create photons from charges that normally don't exist is to accelerate a virtual charge in vacuum (there are virtual particle-antiparticles around, and when there is a (say) black hole horizon nearby it'll capture one of the particle's and the other one is free to accelerate and radiate). This is the so called Hawking black body radiation from a black hole. Still, in some point of view, the charged particle was there. 
At energies above the electroweak symmetry breaking possibly an electroweak charge can do it, create an electroweak field, which when at lower energy would split into electromagnetic and weak. In principle is should, not sure exactly how it would work. Similalrly if grand unification with gravity happens at the Planck scale again some mix including a gravitosomething may do similarly. 
A: I assume that it is a time varying "bent" , which is a gravitational wave.
Only charges radiate electromagnetic waves.
At the elementary particle level photons will be produced only during the interaction time of changing space, i.e. graviton-charged_particle interaction. 
Classically, accelerated and decelerated charges radiate, so a classical gravitational wave will produce electromagnetic waves  from charged matter on its path. 
Now if by "bent" is meant the distortion of space as happened with the LIGO merger of two black holes, then it is gravitational waves that are radiated from masses in general, charged or neutral.
