I have some boxwoods that are really old, some sections are dying

Look on the Internet or visit a local garden shop and start asking questions.

I hadn't trimmed mine in way too long & had to take a lot off @ one time and they did the same thing. Just clip out the dying parts and they will fill in very shortly. Mine look great now. Just don't wait too long between trimmings.

Here is an article that might help you - http://www.southernliving.com/garden/grumpy-gardener/whats-killing-my-boxwood - good luck

I have had many years working for a "Nursery"; sounds like the bush has gone on for a long time stressed. GOOD NEWS... As long as the roots are good & haven't been hurt by a flood then instant freezing or chemical spill, etc. the roots are already massive, to support the old guy. You have 2 choices: cut out the dead limbs first. That is done by "pruning at the joint", snip just above where the branch sprouts from another or the ground (not too close though) & at an angle so the cut sheds water. Next, cut off at the joint any branches that are rubbing on another. To start just look over & cut that 1 branch that really needs it. Then take a good look, wiggle branches if it helps, then cut just that other 1 branch that needs it. Don't try to take in the whole bush at once. You didn't say if the bush is growing freely or has been hedge trimmed just around the outside (the absolute worst thing to do to a bush). The bush is putting in so much energy to keep those inside naked stems alive to support that minuscule shell of leaves around the outside & getting virtually nothing in return for the effort. There is a term for that: "Cauliflowering". If that is the case the plant is so starved for what the leaves aren't doing anymore, it is giving up the ghost. GOOD NEWS... cut the plant to 6" above the ground. Because of the massive root system that has had that crushing burden removed; in a couple of months there will be a puff ball of leaves. The next year the bush will more than double in size, & keep growing quickly & boy will it just look happy.

In the hole put something/anything lawn ornament, bird bath, potted mum or what ever tolerates that sun exposure. I collect free chairs, benches, tables, etc. from "Free Craig's List" to hide/accentuate yard curiosities. If the original planting was too close together & has grown up & the bushes are crowding, cut out every other bush & kill the roots, with a root killer recommended from a Nursery. I need more info to suggest anything else.

Yes, I would trim out the dead ones, and trim the area, so they all will blend well. Dig the area of the trunk of the tree, and place some good soil, there, and then by spring, they will have a good growth for spring of next yr. J.