First, I can't really say what you should do for exercise or eating when you are sleep deprived, but I like LikesLardinMayo's answer. Some thoughts, with studies for your perusal:

(Sleep may beat the extra studying...)

Neurocognitive Consequences of Sleep Deprivation https://www.thieme-connect.com/ejournals/abstract/sin/doi/10.1055/s-2005-867080
Deficits in daytime performance due to sleep loss are experienced universally and associated with a significant social, financial, and human cost. Microsleeps, sleep attacks, and lapses in cognition increase with sleep loss as a function of state instability. Sleep deprivation studies repeatedly show a variable (negative) impact on mood, cognitive performance, and motor function due to an increasing sleep propensity and destabilization of the wake state. Specific neurocognitive domains including executive attention, working memory, and divergent higher cognitive functions are particularly vulnerable to sleep loss. In humans, functional metabolic and neurophysiological studies demonstrate that neural systems involved in executive function (i.e., prefrontal cortex) are more susceptible to sleep deprivation in some individuals than others. Recent chronic partial sleep deprivation experiments, which more closely replicate sleep loss in society, demonstrate that profound neurocognitive deficits accumulate over time in the face of subjective adaptation to the sensation of sleepiness. Sleep deprivation associated with disease-related sleep fragmentation (i.e., sleep apnea and restless legs syndrome) also results in neurocognitive performance decrements similar to those seen in sleep restriction studies. Performance deficits associated with sleep disorders are often viewed as a simple function of disease severity; however, recent experiments suggest that individual vulnerability to sleep loss may play a more critical role than previously thought.

(Sleep is linked to overeating and weight gain)

The metabolic consequences of sleep deprivation
http://www.sciencedirect.com/science/article/pii/S1087079207000202

"The prevalence of diabetes and obesity is increasing at an alarming rate worldwide, and the causes of this pandemic are not fully understood. Chronic sleep curtailment is a behavior that has developed over the past 2–3 decades. Laboratory and epidemiological studies suggest that sleep loss may play a role in the increased prevalence of diabetes and/or obesity. Current data suggest the relationship between sleep restriction, weight gain and diabetes risk may involve at least three pathways: (1) alterations in glucose metabolism; (2) upregulation of appetite; and (3) decreased energy expenditure. The present article reviews the current evidence in support of these three mechanisms that might link short sleep and increased obesity and diabetes risk."

Sleep Curtailment in Healthy Young Men Is Associated with Decreased Leptin Levels, Elevated Ghrelin Levels, and Increased Hunger and Appetite
http://www.annals.org/content/141/11/846.short

Systemic bacterial invasion induced by sleep deprivation http://ajpregu.physiology.org/content/278/4/R905.short
(Okay, that was a rat study, but stil... bacterial invasion is bad, right?)

Short Sleep Duration and Weight Gain: A Systematic Review http://www.nature.com/oby/journal/v16/n3/abs/oby2007118a.html

(Interesting that sleep deprivation is becoming more widespread/chronic - the word "pandemic" is used)

Sleep loss results in an elevation of cortisol levels the next evening. http://psycnet.apa.org/psycinfo/1997-38275-003
(I think Dr. Kruse mentioned this ancient pathway...) ;)

Effect of sleep deprivation on surgeons' dexterity on laparoscopy simulator http://www.thelancet.com/journals/lancet/article/PIIS0140-6736(98)00034-8/fulltext (Speaking of surgeons... Okay, maybe I jumped the shark there.)

Altered brain response to verbal learning following sleep deprivation http://doug06.free.fr/DIU%20Sommeil/Mazza/403655a0.pdf

Okay, maybe you are sleep deprived at 4-6 hours per night, maybe not. What do I know? But if you don't feel fully rested, you could try my strategy for when I'm tired. I take a nap. Usually lasts 20-30 minutes. My cats are really good at this. It's quite refreshing.

sleep is 1/3 of the equation (along with diet and exercise) so rather than trying to figure out a way of side-stepping the issue of sleep, you need to find ways to get the sleep you need. Have you darkened your room properly? Do you do things in your bed besides sleep and be romantic (like study or watch tv)? If so, stop. Are you budgeting enough time for sleep? You need to schedule it, just like you set aside time for your workouts. Alcohol actually lowers the quality of our sleep, btw, so using a night out as a way to sleep in is not a good plan.

Exercise is important so go ahead and exercise. However keep your intense sessions below 45 mins. So you could go for a walk 5 times a week for 30 min (low intensity and would not count against the 45 min limit) and then lift weights. Or you could do a body weight exercise routine in place of lifting.

A few things. Firstly, the eight hours is a recommended length but it's more of a guideline. Some studies have shown that having that much sleep can, in fact, be detrimental and there is a lot of evidence to suggest that you need more or less sleep at different times of your life depending how your body changes. What is more important is the quality of the sleep that you do have. This can be difficult, particularly if you have a lot of things on your mind. Good regular exercise and a healthy diet can contribute to how well you sleep during those hours of sleep that you do get. Try to relax before you go to bed and find a natural way to unwind. Be as comfortable as you can be and don't fight against the sleep. Getting all of the thoughts of the day out of your head is not easy, but the better you prepare yourself for the night the better sleep you will get.