You're speaking my language here. I put 60 loads of wood chips on last year. I use herbicide, but don't like to use it around younger trees, so mulch is a big help in that regard. It does a good job smothering grass and a lot of broad leaves, but know that some weeds will push right up through it.
Plantain, hogweed, sedge, morning glory, and particularly bindweed. I don't know if you have bindweed in SC, but it can be a terrible pest here.
I haven't noticed any nitrogen deficiency using wood chips as mulch. Strangely, I think I see the opposite. No doubt Rayrose is correct that bacteria first draw nitrogen from their environment to break down the wood (later releasing it) but my theory is that since the roots aren't directly exposed to the wood chips, they don't experience that effect. Further, I hypothesize the most broken down particles (richer in nitrogen) are most subject to leaching, and therefore leach down to the root zone during rain, sort of like a compost tea, which may actually add N. This is really conjecture on my part, based on simple observation of the trees.
As Alan has brought up, there is a risk prolonged use can cause excess vigor in trees. This is supported in the literature, but from what I'm seeing this problem may be quickly remedied.
I've had a hard time obtaining enough mulch, so some older trees have had to go without it here at the house. They are the most tolerant of weeds or herbicides, so some of those trees go without. Last season I noticed some of those trees actually didn't put on enough growth, despite lots of mulch applied in previous years. This concerned me enough I thought I was going to have to apply chemical N to some of these trees, but I've since been able to get some mulch on them, so I'm going to hold off on the chemical N for now.
You asked about rodents. My experience is different from Rayrose in that regard as well. It seems to inhibit voles for me. Again, this is theory, but I think voles don't like it as well because, with a good mulch pack, the entrances to the vole tunnels keep collapsing on them. Like us, I think they want to build a house where they can always find the door. At the farm, where I don't have mulch, I have voles.
Like Rayrose, I do get trash in the chips, and this is a problem. I had to quit accepting truck loads of leaves for this reason. A lawn service was delivering leaves, but there was so much trash, I could fill the cab of my pickup with it some days. I also get logs in with the wood chips, but they aren't too bad to sort out.
Wood chips can cause the soil to retain more water than is good for the trees (depending on how much rain and type of soil) but for me the benefits outweigh the costs.