Is it possible to become a neuroscientist without first studying clinical medicine? I am currently in highschool and thinking about what career path I should choose to follow. I am fascinated by the brain, and I would really love to study how it works, or how to model it with computers and mathematics. 

After doing Google searches, I have found that some neuroscience degrees are obtained by studying medicine first and then doing some graduate work on the brain, other websites have told me that neuroscience is a degree in its own right that does not require medicine first. I am concerned about this because I cannot stand the sight of large amounts of blood.

I am just fine with the sight of brains and the way they feel and smell, so I would be capable of doing research directly on them, but without studying medicine. Is there a degree that focuses on the brain and nerve system that does not require treating patients or managing large amounts of blood?
Neuroscience is interdisciplinary! To give you an idea, the neuroscience faculty at my institution (University of S. FL., RU/VH university) generally have Ph.D. in:

Pharmacology

Molecular Biology

Biophysics

Biochemistry

None of them are an M.D. so you have to ask where would medicine come in useful? Ph.D.'s do not generally interact with patients, perform medical procedures or do anything an MD is qualified to do (with respect to humans). Being said, you can have a dual MD/PhD where for example your MD is specialized health science and technology, and your PhD is specialized neuroscience. I think that for neuroscience research involving humans such as drug trials, behavioral neuroscience, etc. this is a preferred route. Some neuroscientists are then experts in computational studies as opposed to wet lab, where a data science background is helpful. 

Some related publications by our faculty:

Hevers W, Hadley SH, Lüddens H and Amin J. Ketamine but not PCP selectively modulates cerebellar GABAA receptors containing a6 and d subunits J. Neuroscience. 28(20) : 5383-5393, 2008.

Segers LS, Nuding SC, Ott MM, Dean JB, Bolser DC, O'Connor R, Morris KF, Lindsey BG. Peripheral chemoreceptors tune inspiratory drive via tonic expiratory neuron hubs in the medullary ventral respiratory column network. Journal of Neurophysiology. 113: 352-368, 2015. http://www.ncbi.nlm.nih.gov/pubmed/25343784

At least with respect to reference one, the methods state results were obtained from rats/mice, from reference two they collected data from adult cats.

You can see that an MD or medical degree isn't necessarily required for a career in neuroscience. Some career postings I can see give preference to those who know some surgical procedures (not just for humans), but at the entry level I'm seeing 3yrs postdoc experience on top of the related PhD, to which you don't have to be embroiled in clinical decision-making and large amounts of blood (maybe on mice, etc.), publication record, etc.

For your intended program, you'll largely want to refer to your university of choice, mind you curriculum and requirements can largely vary by institution!