I received $10,838 of income listed on my 1099-MISC. I am currently in law school and my tuition is $33225. Why does it show I get $0 for the lifetime learning credit?

The Lifetime Learning credit is a non-refundable credit. If you only have income of $10,838, then it is less than the standard deduction of $12,200 and you have no income tax liability which could be reduced by the Lifetime learning credit.

You should a tax liability for self-employment tax that the LLC cannot offset.

and the self-employment tax should be $1532 and as @MinhT1 noted, it can not be offset by the learning credit