Does a sensation exist without a focused mind?

Does a sensation exists when I am sleeping?

I would look for the answer to this question by projecting a sleeping person and asking if any of his sense organs may still be operating on some rudimentary level. My own conclusion is yes, his sense organs are still operating in a rudimentary manner even though he is sound asleep, assuming it is a normal state of sleep (possibly quite deep). The evidence I see is that normally it is possible to wake a person up by shaking him (sense of touch), making noise or speaking to him (sense of hearing), or shining a bright light on his closed eyes (sense of sight), which his eyes would be able to sense through his closed eyelids. I do not see how it would be possible to wake someone up that way if his sense organs weren't operating. (Odors, i.e., sense of smell, often are not enough to wake someone up unless accompanied by some other sensory experience at the same time. And tastes probably wouldn't apply, since a sleeping person normally will not be eating or drinking anything while sleeping -- probably wouldn't be able to do so and might choke or cough if something were put into his mouth by others.)

Is there a sensation of my clothes when my mind is focused on something else (e.g., driving?)

I would say that the sensation exists, specifically, the sense of touch and the action it produces as a result of contact with clothing, but you're not paying any attention to it if your attention is directed toward something else, especially some task as cognitively demanding as driving. The actions of one's sense organs are entirely automatic. The most that one can do, especially with one's sense of sight, is to direct one's attention in a given direction, or close or cover one's eyes.

Does a sensation exists without a focused mind?

"Mind" and "focus" are misleading to use in the context of what sensations are and how the senses respond to stimuli. "Mind" and "focus" tend to imply a conceptual level of consciousness, which is way beyond the level of sensations.

More broadly, there is a brief but very concise selection of excerpts on sensations under the topic of "Sensations" in The Ayn Rand Lexicon. Ayn Rand states the key point as follows:

A sensation is produced by the automatic reaction of a sense organ to a simulus from the outside world; it lasts for the duration of the immediate moment, as long as the simulus lasts and no longer. Sensations are an automatic response, an automatic form of knowledge, which a consciousness can neither seek nor evade.

The next level up is perception, which that same excerpt describes as follows:

A "perception" is a group of sensations automatically retained and integrated by the brain of a living organism, which gives it the ability to be aware, not of single stimuli, but of entities, of things.

Note the essential characteristics involved in perception, compared to sensation: retention and integration. A single sensation cannot be retained or integrated with anything, except in the form of a percept. A sensation is just the immediate action of a sense organ, automatic action within one's faculty of consciousness. Note, also, that perception, too, is automatic, like sensation.