Minimal positive difference of a mulitset of real numbers
Original formulation

(Originally, the OP was interested in the minimum absolute difference rather than the minimum non-zero absolute difference.)

You are asking two different questions. For the first, consider the restricted version in which all numbers are rational, and the decision variant in which the problem is to decide whether your expression is at most some $\delta$. This problem is NP-complete by reduction from PARTITION. (Wlog all numbers are integers, and you can ask whether the difference is at most $1/2$.)

Your other question concerns the decision variant in which the problem is to decide whether your expression is at least some $\delta$. This problem is coNP-complete, and so unless NP=coNP, no nice lower bounds exist (that is, there are no polysize witnesses that the minimal difference is at least $\delta$ which can verified in polytime).

Revision 1

(The question has been changed, and now we are looking for the minimum non-zero absolute difference.)

The problem is NP-complete by reduction from PARTITION. This time assume all numbers are integers, and add to them the additional number $1/3$. Deciding whether the minimal non-zero absolute difference is at most $1/3$ is tantamount to solving PARTITION, and so is NP-complete.