No, not to everyone. It can start at 2,5km, it seem to start to hit me at 3,5km, but I met many who can climbed Mount Kazbek wothout a problem, experienced and newbies. As far as I read, nobody knows how to predict accurately who'll be affected by it, with an exception of people who is living on high altitudes.
Dont think the dog was cold, just chilling:) it was relatively good wrather up there this time and Ive seen dogs around last time when it was harsh and I heard it gets even worse.
Climbing 5km is tough, but if you start with 2.5km+ peaks I think there is an easy way to slowly upgrade to 5km. It is a good way to challenge yourself, you get a lot of rewards. Just try to be as careful as you can.