I was finally able to get past mid-Sept this weekend. I wish I could tell you why it worked, but I really can't. I will explain what I did (worked once, didn't work several other times).
After completing the 4 cup games, the game simulates, I get the 3 teams asking questions, the game simulates, I get drafted, and I hit "Y" to stop the simulation. I saved the game and then manually simulated until 1 Sep. It worked fine. I then simulated 3 days at a time, saving between each. It made it past the middle of September and eventually (don't remember the date) made it to the preseason games. So far, that Live the Life character is working as it should. My son made it through the preseason over the weekend and is ready start the regular season.
I'd also request this not be marked as "solved." Nothing is really solved. I built a character and tried to do the same procedures several times. The game froze up in mid-September each time. Could it depend on the team that drafts you? My son was on the Lightning when it worked. He was originally on the Ducks when he couldn't get through. I was on the Bruins and still haven't gotten my character to work. The other difference is that he was drafted first overall. We were both drafted 2d in the games that didn't work.
I hope this helps someone. Typos and limitations of the mode aside, this is my favorite part of NHL 14. There's nothing quite like seeing the look on your 9-year old's face when he got "his" face on a billboard!