The site for the cup matches is random, so playing 3 of 4 on the road is bad luck - though BB could take previous games into account. For example they could put the teams with the most road games in one bucket and with the fewest in another bucket and then draw one team from each for each game.
The home team will win more games, so even with the above method it is likely that some teams will start with two home games, but it would be rare to start with two away games.
You missed the first league game, so actually your team has played 3 league games at home rather than 2.
The league season is divided into 3 parts.
Part I is 7 games within the conference, so 4 teams have to play 4 at home, a 4 have to play 4 on the road. It also is necessary that 3 of the teams play back-to-back on the road (as you did for games 4 and 5), and 3 will play back-to-back at home.
Part II is 8 games against teams in the other conference, 4 at home and 4 away. It appears that all Big 8 teams played away for the first game of inter-conference play. While this isn't necessary, it does simplify scheduling, and does provide a certain balance. For example, you know that all the other Big 8 teams have played one away inter-conference game.
Part III is 7 games within the conference, playing the opponents in the same order as for Part I but the opposite location.
I'm pretty sure that the schedule could be improved so there are not teams playing 3 league games on the road in the row, nor finishing with two home games.