What is the ruling on using materials that contain alcohol in their composition, such as soap, shampoo, and air fresheners? How can one guard against them? What should be done if these materials come into contact with bedding or clothing? Is it necessary to avoid others if they use them? What about their presence in the mosque?

It appears your question stems from obsessive thoughts (waswasa), and you should disregard them. The impurity of alcohol (khamr (definition)) is a matter of scholarly disagreement, and you may adopt the view of its purity if the matter becomes difficult for you. You are not obliged to inquire about the use of alcohol in products, as the default assumption is its absence.

Some scholars have held that if khamr is mixed with other substances in such a way that its effect is not apparent, then it does not cause impurity. This was the chosen view of Sheikh Ibn Uthaymeen.

So, the matter is straightforward. The default assumption is the absence of alcohol in products. And even if it were mixed in a way that does not affect them, you may adopt the view of those who hold that liquids do not become impure unless their characteristics change. If, hypothetically, alcohol were mixed in a large proportion, some scholars maintain the purity of khamr, and you may adopt their opinion, especially if you are prone to obsessive thoughts.