Entanglement is a prediction of quantum mechanics, and all experiments on it so far have confirmed quantum mechanical predictions. However, none of them have required something you do somewhere to "affect the other instantly". The two offending words are "affect" and "instantly". An affect implies causation, and causality is limited to the speed of light (no entanglement experiment violates that). Also, "instantly" is a frame-dependent concept (a la relativity), and no entanglement experiment violates that either. What happens is, an experiment you conduct on one electron gives you information about the other-- information that no one else has and no one else needs to successfully use quantum mechanics to make their own predictions. The only way you see this as an "affect" is if you apply philosophical concepts that are outside of science, like if you say that the electron "has a wave function" and changes in the way you would describe its wave function somehow actually affect the electron. The fact is, the only time we ever know "the wave function" of anything is when we prepare a system to have a certain wave function, thereby erasing whatever history it may have had. When we don't do that, as we don't in entanglement experiments, then we only know the wave function that we are using to make our prediction for what will happen-- and someone else with different information will use a different wave function and will find that quantum mechanics works for them too.
When stated in a proper way, entanglement is experimentally verified. Unfortunately, there is a whole lot of baloney written about this effect, ranging from unnecessary philosophy to just plain wrong physical claims.
You don't create them necessarily, it's fine to use pre-existing ones. They can be entangled based on their history. But you are completely right-- it is quite possible, depending on the experiment, that they will also be entangled with other electrons in ways that don't matter to the experiment and we simply choose not to include in their wave function. That's the key point.