All of the above are the body using various methods to expel or combat the virus. They're similar for many viruses because those are the tools our body has to fight viral infections (though obviously it's not all the tools)

Strictly speaking, it's not the virus causing the fever or the sniffles and sneezing. Your body does these in response to the viral or bacterial threat. Raise body temperature in an effort to literally cook the invader to death. Sneeze to expel the invader, etc.

We only have so many symptoms to go around because our bodies can only respond in so many ways.

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The infections could cause all sorts of problems.

Your body's response will be fairly similar in many cases: "intruder alert, expel everything!" is a common one and covers sneezing, coughing, vomiting, and diarrhea, depending where the infection is. Likewise, your body sends a squad of white blood cells in full riot gear to the site of any infection - which then causes swelling as they flood the area.

the symptoms are always the same because the symptoms are not about the infection; the symptoms are the body's reaction to the infection

the things you listed are how the body fights infection

the infection can vary, but since it's always the same body, it'll always be the same method of dispelling infection

The symptoms you feel are caused by your body fighting the viruses. Fever is your body heating up to try to overheat and kill the virus. Vomit/Diarrhea is your body getting rid of whatever you ate via the nearest exit, mouth/butt. Snot, sneezing, etc is your sinus creating a mucus layer of protection to try and catch and remove the illness. Coughing is when that mucus makes it into your lungs.

Because a lot of the symptoms of common viral and bacterial illnesses are not caused by the infection, they're caused by our immune system ramping up to fight the infection. Many of them even serve a constructive purpose, or are side effects of stuff that serves a constructive purpose. Examples:

• Fever - turning up the thermostat can inhibit reproduction in some pathogens
• Congestion - increased mucus production protects mucus membranes in the respiratory tract and transports debris (dead cells, mostly) away from the respiratory tract faster. Sore throat is mostly a side effect of mucus draining down the back of your throat (into your esophagus, and the extra mucus in the stomach can cause upset stomach, or at least diminished appetite).
• Fatigue and body aches - A Defcon-1 immune system is energy-intensive. You're tired because you're spending that energy, but also because the fatigue signals conscious you to stop doing stuff and go rest so your body can spend its energy fighting the infection.

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Plus stuff like sneezing, coughing, and diarrhea are useful for getting new bacteria and viruses out of the body and into another host. So germs that cause those kinds of reactions have a selective advantage.

Many of the symptoms that we associate with being sick are not actually directly caused by viruses or bacteria. Our immune system can respond in both general and specific ways and many of the “symptoms” we get are really just general immune responses. One example of this is fever! Many different viruses, bacteria, parasites etc. cause us to develop a fever. A fever is nothing more than your immune system cranking up your internal temperature so that you can essentially “melt” the proteins that the pathogen is creating. In fact, it is not that uncommon for your immune response to be the direct cause of death in severe cases of infection. Your immune system doesn’t know that by giving you a fever it could melt your proteins as well as the invading proteins. All it knows is invader=raise body temperature. Another reason many different pathogens cause the same symptoms, is that there are only so many pathways a pathogen can take to infect you. If you breathe in a water droplet with some virus on it, you will most likely develop respiratory symptoms (ie. sneezing, coughing etc.) because that’s just the part of your body that is fighting back. Similarly, if you eat food contaminated with a pathogen you will most likely experience diarrhea, vomiting etc. because that is the part of your body fighting back!

The easiest most simple explanation is that your body will turn up the heat to slow pathogens. The more you sleep the more energy conservation your body has to fight pathogens. Your body will use different nutrients to fight the bacteria. Your body has a type of cell called leukocyte that patrol the blood being on the lookout for anything abnormal. When depleted, you need protein to help replenish. That's why chicken noodle soup is so great. It's easy on the digestion system and has protein in it. Drinking lots of water helps clear our the kidney and liver allowing you to expel the pathogens.