I hate to be the party pooper but the Ibex video is not real. I wish it was real but it isn't.
To the question 'Do they fall?'. Yes they do. I've found their remains at the base of cliffs and in an avalanche path. Surprisingly many of these falls are not accidental. Mt Goats can be pretty aggressive with each other. I have some fighting photos on my hard drive, I'll try to load them tonight.
Here is one of a goat that fell and survived with both horns broken.
I like to think of them as my 'Alpine Angels', but they don't always act very angelic.
One of the best areas i know of to observe both rocky mtn bighorn sheep and mtn goats is the Ruby mtns of N eastern NV. Spend a week backpacking the ruby domes areas and youll probably see some exciting animal climbers.
I have seen their hair caught on small bushes along 5.8 climbs. I was amazed to see their tracks in a light covering of snow over ice on a 5.7 slab. Also saw a baby goat leap down a 10 foot cliff short-cutting to get to his mother.