I thought when talking to one person informally, the stem of the "du" verb is used. Am I wrong? The du conjugation has an umlaut. Is the rule that you use the "ich" form of the verb? I'm confused.
What you said is true in the present tense, but this is imperative. The informal singular imperative form of "lassen" is "lass" or "lasse" (interchangeably I believe), and the pronoun "du" isn't used, unlike the formal imperative where "Sie" is used.
Check out the Imperativ section here for more information: http://conjugator.reverso.net/conjugation-german-verb-lassen.html
Lass mich in ruhig makes no sense.
You're probably thinking of Lass mich in Ruhe, which is something like "don't bother me" -- it doesn't necessarily mean "leave the room; I want to be the only person here" but just "I don't want anyone bothering me or intruding on my attention".
So it means "leave me alone" in the sense of "stop pestering me" but not in the sense of "cause me to be alone in this area".