A few people, either though comments or E-Mail have mentioned some issues. I've replied directly, but decided to also address them here.
Bare Feet - yes, most people would be hobbling after a while in those conditions. A couple of them don't have the issue, having spent enough time barefoot back home to have calluses. Some type of sandal will be made for those who need them.
Placement - one person mentioned that Saber-tooth existed in North America, so that's where they have to be. While Smilodon and a few other species were N.A. native - other species existed in Europe, Asia, Africa and South America - so that's not going to help. I will say there's a clue in Chapter 2 for the careful reader, but I won't spoil that.
Time - They are in a warm period, but it's not the Holocene - the current warm period. Glacial periods happen in cycles and they could be in any previous warm period - how they figure out when will be taken care of in a later chapter.
I'm not just writing this on the fly, I've done a good bit of research to make sure I'm not turning this into fantasy.