What exactly is charge? If gravity is really the bending of space/time causing objects with mass to experience acceleration, is there a similar physical meaning to 'charge' besides 'a property of matter which causes it to experience a force when placed in an electric field.'  What exactly is charge?
 A: What charge isn't ( or doesn't seem to be so far) is something that has properties similar to gravitation in that its fundamental effect is based on curving space-time, as in general relativity. There have been theoretical attempts to formulate a theory of electromagnetism that resembles general relativity (such as the Kaluza-Klein theory  and other more recent theories along similar lines).
But so far, such theories have not been verified with experimental observations, such as the particle physics experiments done at CERN. That's not to say they never will be , just that so far there is not sufficient experimental evidence to say charge works like gravitation in this way.
The theory that so far seems to have the strongest experimental verification is quantum electrodynamics (QED), in which the charge is a quantized property of matter (i.e charge does not come in any smaller units than the charge of an electron or positron, with the exception of quarks, which have one-third or two-thirds the charge of an electron or positron). The electromagnetic force in QED arises from the exchange of virtual photons between charged particles. 
It's hard to give a better qualitative explanation within the scope of the question without digging further into QED (and that would entail some deep digging), but hopefully this is helpful.
A: Well what is mass in this case? Just because you can see in the macroscopic world the effect of mass doesn't mean what you think of it is tangible. Its the same thing with charge. Charge is just an intrinsic quantum number that nature chooses to interact with through the EM field. The same with mass, wherever "mass" comes from (vacuum energy, higgs mechanism etc) it is really a property which defines the coupling strength of gravity. It equally as abstract as charge is, but we are way more familiar with mass in our everyday lives.
A: This is a personal oppinion, not available in textbooks. AFAIK there is no answer to that question.
An electron collides with a positron, both have charges and the resulting gamma rays (light) do not have charge.
Your question can be upgraded to the one: Light is charged ?
I'm tempted to say yes. Light has both charges at the same time.
The inverse process is also valid : two gamma rays collide giving an electron and a positron.
As seen, charge is present before light was created and also after it was destroyed. In between, in the light phase, both charges should be there.
What is charge? I dont know, yet. One operational answer can be: charge is the origin of the electromagnetic field. 
(more than what you said: it responds to an electric field)  
