LED's shouldn't be a problem just remember they are polarity sensitive. I'm assuming the system is 12v correct?
Turn signal not flashing could be a few things, a common one is that some kits don't come with a flasher unit, if the system is wired without one you will have solid lights.
I would fuse the lighter but the one in my truck is not fused and that looks factory.
For the license plate light I would check grounds first.
