Homemade Butter and Cultured Buttermilk from Heavy Cream

Yes. When you start with raw cream tp make butter, then your byproduct of buttermilk will already be cultured. No need to add cultures since raw milk already has those active live cultures known as probiotics. Note: since raw milk is not pasteurized, there can be harmful bacteria present in additional to the good bacteria — the probiotics. So drink raw milk at your own risk.
Depending on what state you live in, you may be able to find a small market or local dairy farm that sells raw milk if you're interested in getting the health benefits of raw milk and are not afraid of the risks. Read more about the risks of consuming raw milk here.

You can use a churn, a blender, a food process or a mixer. The taste will be the same. An old fashioned butter churn can be a fun activity to do with kids whereas modern tools offer speed and convenience.
Believe it or not, you can actually purchase butter churns on Amazon.
If you're trying to just get some homemade butter — stat — though, you might just want to go ahead and use a stand mixer or a food processor.

Yes. Cheesecloth is extremely helpful for making homemade butter. Homemade butter needs to be rinsed to remove liquids that can spoil and cause your butter to taste rancid. Cheesecloth acts as a strainer, containing the milk fat while rinsing away liquid with cold water. You can buy cheesecloth at some groceries stores or online. Shop cheesecloth on Amazon.

Homemade butter does not need to be refrigerated as long as the milk is completely removed from the butterfat. However, I recommend keeping your butter refrigerated just in case some of the milk remains even after you thoroughly rinse your butter. The milk can spoil and can make your butter taste rancid.

Yes. Cultured buttermilk can spoil if left at room temperature for too long.
During the culturing process it may need to be exposed to warmer temperatures. Refer to the instructions on the culture starter package for this.
However, once the culturing process has been completed, you will need to refridgerate to prevent buttermilk from spoiling.
Buttermilk should not have a foul smell. If it does, it's likely spoiled and should be thrown away.

Basically, bad things. You will want to take a little extra time to make sure you squeeze all the buttermilk out and thoroughly rinse your butterfat with cold water. This mean repeating the rinse and squeeze process as many times as needed for the water to run clear.
Why? Well, first, you don't want your butter to be milky because this will make things like your toast soggy instead of nice and crispy.
Second, any leftover buttermilk will start to go bad quickly, giving your butter a rancid taste. You butter can potentially become contaminated with bacteria as well.

Yes. But it will not be cultured butter.

When heavy cream is shaken, stirred or churned, the fat separates from the milk. The reason this separation occurs is because the membranes surrounding the fat in milk are very fragile and they rupture when agitated (source). Once the membranes rupture, the fat can attach to other fat and eventually begins to form clumps, leaving behind the milk. This allow you to separate the milk fat, add salt to it and call it butter!

To make cultured homemade butter from pasteurized heavy cream, you'll need add cultures to the heavy cream, before you begin the butter-making process.
To do this, you can add culture starter or plain Kefir to your heavy cream. Follow directions on the package for the culture starter.
If using Kefir, add 1 tablespoon to your pasteurized heavy cream, cover with a towel or loosely with plastic wrap or a lid (so that it's not airtight), and let the cream sit overnight at room temperature or until cream becomes thick.
Note: If you notice a foul smell, toss the heavy cream. It's likely spoiled.

For food safety, commercial milk goes through a pasteurization process to kill harmful bacteria. However, the pasteurization process also kills active cultures that contain helpful bacteria. You've probably heard them called pro-biotics. There are a number of drinks now you can buy that have been re-cultured after pasteurization in order to re-gain those pro-biotics.

This is a popular hack if you don't want to deal with culturing. The flavor is produces is similar to buttermilk, but it's not true buttermilk. True buttermilk goes through a chemical process and produces a different taste and a thicker texture. You also get those healthy pro-biotics when you culture the milk using actual cultures. If you're really after the authentic flavor and smooth, creamy texture of buttermilk, then culturing the milk leftover from making butter, is the way to go.

When you culture milk, the fermentation process actually transforms lactose (the sugars found in the milk), into lactic acid. Lactic acid is much easier to digest than lactose, which is appealing if you are lactose intolerant.