If your Christmas cactus is dropping branches, there could be multiple reasons. Aged specimens self prune, for example.  If the branches that are dropping (leaves) are flat instead of plump and hydrated, it isn't getting enough water pumped up from the roots to circulate through the plants system and it can't support the upper growth, so it drops them. This could be from overwatering and lack of air in the bottom of the pot or from not watering enough. Of course, buds could drop if you've moved a plant to a noew location too. 

These plants are epiphytes that are frequently found in trees in a tropical environment that also gets a very dry period. So, they like to get more water than desert type succulents, but they need very good drainage. The planting medium should hold enough water to sustain the plant, but also be shedding the excess so the roots aren't standing in water. In the fall, they should be watered less than the rest of the year. If you were to examine the roots of your mother plant, they should be quite white and firm and not mushy, brown.

If you take some fallen leaves, the ones to propagate most easily are those that are still plump and firm. However, if all you've got are dehydrated ones recently fallen, let them callus at the attachment point for a day or two then put just the tip into a container with a small amount of water. The leaves should gradually plump up by drawing in water and slowly you will see some root formation in that bit of water that will help you form a new plant. 

As for the woody portions, yes they can also be propagated but it takes longer. You will need to use a rooting hormone on these and use a succulent type potting mix for the medium, that is moistened. Make a clean cut, dip the end and shake off the excess and insert it into the mix. Make a mini greenhouse such as using a baggy/ziplock that doesn't touch the plant but keeps the humidity in.  You will want to keep that soil mix slightly moist. Eventually that woody cutting should produce roots. The woodiness is normal as the plant develops that to support the heavy growth it produces over the years. 

If the soil has drawn away from the side of the pot and it frequently looks this way, you are underwatering. If the soil at the bottom is soggy and the top is dry, you may need a better pot and to do a soil change-repot. Many people don't like to do this with big old plants like Christmas Cacti or Jade plants because it is awkward to handle so doesn't get done often. 

Yes, that kind of soil for succulents often has enough peat in it to be more moist than desert succulents want but OK for tropical type succulents. 

A pot with enough drainage and perhaps wider than deep.