Through every pregnancy I've caught everything going and this one is no different other than getting the flu jab and not catching flu! Your immune system is weaker so it could well be a nasty cold. I wouldn't worry as long as you don't have a high temperature and stay hydrated. I doubt it's because of the flu jab as it isn't a live vaccine, so it won't actually cause flu like symptoms.
I don't know really. I have had sneezing through out my pregnancy but when I got the flu shot it seemed to get worse, and I did get a runny nose bad. That all being said, even if that happens, I still believe getting a flu shot is better then not getting one.