If your soil is brick-like, you'll likely need to soak the whole rootball for a while, to help soften up that old soil so you can break it all apart & crumble off every last bit of it. The plant will be fine, they're pretty tough survivors even tho' they may not look it. I make a mix of roughly 1/3 each Potting (or African Violet) soil, Perlite & ReptiBark (from the Pet Store), drains well to allow for deep watering. I'd repot it all dry, wait a day & then water well.

For large plants with stiff spines I wrap a piece of carpet or a towel around the cactus and hold the ends of the towel/carpet together to maneuver the plant around without getting poked. It's sort of like having a big, soft pair of pliers.

I always repot this type of cactus immediately. Check to see if it has roots, although it doesn't much matter because by the time you get the concrete ball of peat off there won't be many roots left! Also check for pests. I take the opportunity to soak the plant in soapy water for an hour or two which kills most things. Then plant into a dry mix. Leave for at least a week before watering, possibly longer if you destroyed most of the roots.

Christmas cactus (or any of the holiday cacti, for that matter) touchy??? Are we talking about the same plant(s)? I grow many Schlumbergera x buckleyi (Christmas Cactus) and Schlumbergera x truncata (Thanksgiving Cactus, often mistakenly called Christmas Cactus), and find them some of the easiest plants to grow that there is. Transplanting them should be very easy, and I doubt the process will even slow them down. True Christmas Cactus does tend to be more weeping than most other holiday cacti. I doubt you could change this to any degree by doing anything at transplant. One thing you could try is keeping new growth in check by pinching/removing new segments while keeping the older segments supported until they are more mature (firmer). I've used bamboo skewers for this purpose with decent results. Another thing that helps is selective pruning. Removal of the more weeping growth and keeping the more upright growth can lead to a more upright plant. Keep in mind that the plant's natural habit is weeping/low-growing, so you are fighting nature (an uphill climb) when trying to change that form significantly.