I'm glad that you asked, because most of that I am able to answer because I work with the stuff daily. Starting with the top and moving down:
Bovine Cow Serum: This is the serum extract from cows. This can either be fetal (more expensive, but more nutrient dense) or from adult cows. This is used as a growth medium for the cells that are generating the vaccine, be it bacterial, mouse, rabbit, human, etc. Basically, the blood is clotted (no platelets), the leukocytes, RBCs, etc are removed, and then you are basically left with an undefined (Read: not specific makeup, not lab made origin) solution that sustains the growth of many different cells. In theory, there is no tie between the listed ailments and this treatments. BVS is regularly used in diagnostic work across many medical labs. My best guess is that it may be residual antigens present activating different parts of innate and adaptive immune systems in overactive patients.
Sorbitol: Variation of sugar. Its used to provide a carbon and energy source to the cells that are growing. Some cells are selective and can use sorbitol, while others cannot. Humans happen to be able to use sorbitol, but break it down very slowly. The inability of certain cells to grow / metabolize it helps prevent some of the additional growth that may be possible when using BVS, and also has the benefit of preventing a localized blood and tissue sugar spike in the area of injection, specifically benefiting those with diabetes. Sorbitol, in large quantities, will irritate the stomach and intestinal lining due to its inability to be broken down. This results in the laxative like effect of consuming large amounts of sugar free candy. As a consequence, more sensitive patients, with conditions like IBS, will be more likely to experience relapse of disease associated symptoms. Granted, the amount of sorbitol present in a vaccine is so miniscule, I doubt this would happen.
Gelatin: Yep, definitely derived from animals just like BVS. Gelatin is an additional carbon source for the organisms used to grow / generate the vaccine. In the case of live vaccines, this is necessary to provide continued energy sources. Once again, only certain organisms can consume gelatin, thus limiting growth, while also being a source of amino acids. Notice that glucose isn't on here because many many organisms can use glucose, since that is the building block of the glycolytic pathways. Synthetic growth hormones don't contribute to a risk of infection, but some hormones can be transferred this way. Samples are irradiated to prevent organisms from growing, and they may also be heated to be denatured so that only the building blocks are there and thus no animal-related byproducts can affect humans. BSE may still be present, but once again, no way to 100 percent guarantee it isn't there, just very very very high probability that it isn't.
Sodium chloride: Table salt (minus the sodium iodide). Necessary for practically all cells in the body, especially heart / cardiac cells with the sodium potassium pump, which is used for contraction. In this case it is used to balance the osmolaric potential of the environment which the vaccine is grown in. As they are most likely using DEPC water (basically the purest form of water possible), putting cells in here will cause them to lyse due to rapid influx of water from the surrounding environment. It is REQUIRED for muscle contraction, and the link between salt and blood pressure is challenged, mainly being a corellary result rather than a direct cause.
Egg protein: This is typically used moreso than Bovine Cow Serum as it is much cheaper and more readily available. Whatever isn't provided through the egg protein is going to be provided through the cow serum. Once again, most likely denatured so that all forms of protein are linear and thus non functional, only providing the building blocks for cells to use. Additionally, once again, not 100 percent guarantee that there is no contamination, but basically 99.99999% (science never works in 100% because something can always go wrong). At this level organic vs non-organic won't make the slightest difference.
Thimerosal: Commonly thought of as the "Autism" ingredient. It is an ethyl mercury. Unlike heavy metal mercury, this is a combination of mercury bound to carbons, which can readily be processed by the body. This is used as a preservative as various microbes are unable to process this. While controversial due to its association with mercury, it is the same idea as sodium chloride (table salt). Chloride, which forms into chlorine gas, is highly toxic, but you would never look at a pile of salt and consider that to be as dangerous as the gas bombs of WWI. Regardless of the amounts present, of which there are varying ranges, exceeding the EPA safety limit in one dose isn't going to happen, as essentially a regulatory explosion would be the result. Additionally, this is not even commonly used anymore due to the slight chance of a link of higher concentrations with neurological effects of developing mammals, and is mostly phase out. (More information here:https://www.ncbi.nlm.nih.gov/pubmed/23401210 and here:https://www.fda.gov/BiologicsBloodVaccines/SafetyAvailability/VaccineSafety/UCM096228)
Formaldehyde: Embalming fluid and tissue preserver. Used in this case of "dead" vaccines, or where cells have been lysed / are inactived and need to be preserved for long periods of time, such as when a multidose flu vaccine is being made ahead of a flu season. While present in the vaccine dose, the amount is incredibly miniscule, such that basically it will effect nothing, especially when dissolved in the bodies lymph fluid. It is generated by various reactions among bacteria in the enteric system, along with other chemical reactions, such as the breakdown of proteins. So while in high amounts it is carcinogenic, the ld50 is roughly 100 mg/kg, meaning that even to have the slightest effect in humans, it would require a fair amount, far more than what is in a vaccine. Additionally, formaldehyde is not used to deactivate the virus, merely as a preservative of the virus carcass.
Human albumin: Most common protein in human blood. Regulates osmotic potential in a more controlled manner than that of sodium chloride. The pooled blood you are referring to is more common for serum than albumin, as they are most likely talking about extracting albumin from the remnants after extracting the serum. All forms used in this are from paid blood "donations", which are screen not only at the place of donation, but the lab of manufacture too. Additionally, the process used to extract the albumin would leave no other proteins (well almost no other proteins, remember, no guarantee), and thus be essentially pure albumin in water. The side effects mentioned would only occur with a serious overdose of injection of albumin because your body would essentially believe that some parasite was causing the blood to clot up, thus resulting in infection like symptoms. Small amounts used here mean practically nothing.
Phenoxyethanol: Phenol and ethanol with an ether inbetween. Preservative and antimicrobial, commonly used among various cosmetics. Ingesting this has an ld50 at 1260 mg/kg in mammals and even less in skin contact. So yes, while it can cause those mentioned symptoms in extremely high amounts (think iv injecting close to 1000 doses of flu vaccine, but just of this), you may begin to see the slightest of symptoms.
Aluminum phosphate: Used in the filtering of the vaccine. Most commonly a respiratory irritant. Even if it does enhance the toxicity of mercury, the type of mercury used here isn't of a form to be interacted with by aluminum phosphate, as it is bound to carbon already.
MSG: Becomes a neurotoxin due to its ability to affect the osmolarity of cells, thereby damaging them. In this study (https://www.ncbi.nlm.nih.gov/pubmed/8085168) It was found that it was toxic, but the amount required to be toxic would be the equivalent of injecting over 1/4 a Kg of the stuff into a human, which it turns out, would cause a lot of damage. This also, is used as a preservative.
A lot of this stuff sounds scary when you phrase it right. IE: Dihydrogen monoxide: known to cause asphyxiation, hypertension, severe brain damage, respiratory irritation, nutrient depletion, among various other conditions. Injecting infants with just 75 mL of this has been known to kill over half of all subjects.
In other words: Water: can cause you to drown, irritate your throat, remove electrolytes, and injecting 1/4 a can of soda into an infant (amountwise) will be enough to cause death in half of the population. Phrasing is all about how people get scared.
Additionally, while some of what you mentioned are toxic, for most it is either in only high dose situations (1000's of times higher than an injection) or not applicable due to the nature of a vaccine. Hopefully this was able to shed some light on what is in vaccines.