Our three-legged cat Sammi is just as active as any four-legged cat. Soon after adopting her we realized she didn't have any limitations climbing up a perch or jumping onto the couch or bed. Her difficulty came in getting down from a multi-level perch or anything more than about 2-3 feet high. The standard perch levels were too far apart and too vertical for her to navigate down safely. She would tip rump-over head because her inertia would carry her rump down quickly and she couldn't absorb all that momentum on one front leg. She seemed to "fall" off the perch as often as she "climbed down" it.
To make things safer and more fun for her we bought more perches and adjusted the heights so she could go criss-cross from one perch to another. The photo shows we put the perch on the left up on cinder blocks and it became the perfect height for her to climb up and down. It took only a few times of us showing her the easy way up and down before she quickly learned the new path and happily romped up and down her new perches.
There are only two things we've done to help Sammi live an easier life with three legs. We've made sure there is an easy way down from every higher-up location she visits. We put a lower step beneath every window-seat so she needs to jump only about 2 feet down at every level. A one-level perch, ottoman, or upside-down sturdy wicker basket have all worked well for this purpose. The other thing is to use uncovered litter boxes and not make the litter too deep. This helps her get good footing in the litter box and be able to get in & out from any direction so she doesn't have to turn around on the litter that shifts underfoot.
Sammi plays well with Raven, who we adopted at the same time. Being 10 years old, Sammi is less inclined to play than 4-year old Raven, but her 3-legged status doesn't appear to be a factor in their romping and chasing. Raven likes to ambush Sammi and they both playfully bop each other on the head – Raven with either front paw and Sammi with her one paw.