If the disposition of a county court case is DISM (dismissed) was the defendant convicted of the crime?

When the probation is over, the case is "dismissed". You are right - this is confusing because the charge itself was not dismissed and if the person received probation, then it remains on his/her record (although they may later be able to petition to seal the record.)

It may be that the conviction was deferred. Often times, a person is placed on a probation called 'deferred adjudication.' What that means is, the judge finds that there is sufficient evidence to find a person guilty, but delays that finding while the person is on probation. If the person completes the probation satisfactorily, the judge dismisses the charge without a finding of guilt. A person has to plead guilty or no contest to receive this sort of probation.