1. No, the NR background model is built into the source code. There isn't any code in FIMO that accesses the network.
2. FIMO and most of the rest of the MEME Suite tools only use a 0th order background model. MEME is the only tool in the MEME Suite that can use higher order models.
3. Ideally the background model should be derived from sequences that are biologically similar to the sequences you are scanning, but that don't contain instances of the motif you are scanning for. If you are using FIMO to scan selected enhancer regions you could try generating a background from unrelated enhancer regions for the same organism. If you are scanning all of hg19, then you should probably generate the background from all of hg19.
4. Providing an appropriate background model is the single most important step you can take to improve your FIMO results. The FIMO match score is the log-odds ratio of the probability of observing a sequence given the motif PWM to the probability of observing the same sequence given the background model.