There used to be a tab under Challenges where there are BeginZ challenges listed (this is way back when BG BeginZ was released for NA/EU).
There were 6 of them in total:
- Win/Complete BG BeginZ in Easy
- Win/Complete BG BeginZ in Normal
- Win/Complete BG BeginZ in Extreme
- Wipeout 100 bots in BG BeginZ Easy
- Wipeout 300 bots in BG BeginZ Normal
- Wipeout 500 bots in BG BeginZ Extreme
In short, for every game you successfully complete, you get some reward (not a lot). For every 100/300/500 bots you kill, you get whatever rewards you could get depending on the difficulty. It is speculated that you get more chances of getting the perm weapons if you complete these challenges.
Since that tab was removed (you can no longer see the Events tab), people still firmly believe that this reward system was never changed.
As for the solo playing, I have to guess it's to prevent people from farming the mode. Don't take my word for it. Maybe it was there since its release.