What senses are active while sleeping?
The answer is of interest not only in sleep but also the perceptions of patients under anesthesia, comatose states, etc.
Our senses aren't 'dimmed' in sleep. There is no effective way to turn off our senses. The best way to explain what happens in sleep is that at some point (the last point, actually), our cognitive processing of sensations changes. That is, our higher brain functions allow us to ignore certain sensory input. This effect has been most studied in sound perception.
In a sleep study involving sound and EEG/fMRI during all phases of (drug-free) sleep,

The first significant result of this study is that the pattern of brain activation associated with auditory stimulation was strikingly similar in wakefulness and sleep, suggesting that sensory processing occurred in both conditions. However, we found qualitative differences in brain activation associated with auditory processing during sleep compared to wakefulness. The reduced regional activity during sleep, compared to wakefulness, in the left parietal and, bilaterally, in the prefrontal cortex, thalamus, and cingulate gyrus (part of the limbic system) suggests that these areas may be involved in the further processing and perceptual integration of sensory inputs likely to occur during wakefulness only.[1]

Supporting this is evidence that some sounds awaken someone no matter at what stage they are in sleep, for example, young mothers are woken up by their infants' lightest movements[2], and the fact that we respond to our own names in a similar fashion in sleep and in wakefulness, but such responses are not shown for presentation of other first names.[3]

These results suggest that the sleeping brain, during SII and PS, elicits a differential cognitive response to the presentation of the subject's own name, comparable to that occurring during wakefulness, and therefore that the sleeping brain is able to detect and categorize some particular aspects of stimulus significance.[3]
Thus, the results suggest that when subjects were listening to their own name during sleep some brain regions were selectively more responsive than in any other condition. ...we have demonstrated that the sleeping brain is able to process auditory stimuli. In addition, we postulate the existence of a functional network capable of detecting and facilitating processing of emotionally relevant inputs during sleep.[1]
Our results support the view that PS is not a state of &quot;sensory isolation&quot;; failure to respond to external stimuli during this stage may depend upon mechanisms occurring only after the sensory input has undergone cognitive analysis.[4]

However, as you noted, odors are not good at waking people up.

Some suggest that the human olfactory system during sleep is sufficiently well tuned to ensure arousal to such threatening stimuli as odors associated with smoke from fire. Our results strongly suggest otherwise. The intensity, strength, and noxiousness of the pyridine stimulus elicited behavioral arousal or EEG activation on fewer than half of stage 2 trials, less than one third of REM-sleep trials, and virtually no stage-4- sleep trials. This is a nontrivial lack of response, since pyridine is a component of coal tar and is also used as a herbicide for firewood, and thus is a likely by-product of many real fires. In practical terms, therefore, olfactory awareness in humans is low to absent during sleep, and human olfaction appears insufficiently sensitive and reliable to act as a sentinel system. We further note that auditory arousal threshold is highest in young and sleep-deprived individuals, increasing the likelihood that olfactory processing is even worse for children and sleep-deprived adults.[5]

The methodology in the olfactory study varies quite significantly from the auditory study. Better studies may shed more evidence on why olfactory stimuli are processed differently from other stimuli.
As to what percent? That probably differs from person to person, and I did not search the literature for that.
[1] Auditory Processing across the Sleep-Wake Cycle: Simultaneous EEG and fMRI Monitoring in Humans
[2] Changes in sleep patterns of young women from late pregnancy to postpartum: relationships to their infants' movements
[3] A differential brain response to the subject's own name persist during sleep
[4] Brain Processing of Stimulus Deviance During Slow-Wave and Paradoxical Sleep: A Study of Human Auditory Evoked Responses Using the Oddball Paradigm.
[5] Minimal Olfactory Perception During Sleep: Why Odor Alarms Will Not Work for Humans