Can the droppings of dogs and mice be considered pure based on Fatwa No. 168078, and does the ruling on the purity of donkey dung apply to them?

It appears you have some waswas (obsessive whisperings/doubts), and it is advisable to disregard them. Below are the answers to your questions:

1. The vacuum cleaner used to remove traces of the dog is of doubtful impurity and remains pure according to the original state of things, which is purity. Similarly, sports equipment does not become impure merely by doubting its contamination with dog feces or mouse droppings, because the original state of things is purity until impurity is established with certainty.

2. It is advisable to ease your mind from doubts about impurity. You may use sports equipment and other items, and it is not necessary to wash your hands after exercise merely due to doubt about their impurity.

3. It is permissible to follow the opinion of purity regarding the waste of animals whose meat is not eaten. This is an opinion held by some respected scholars. A common Muslim is permitted to follow a scholar whose knowledge and religiosity are trusted, and is not obliged to adhere to a specific madhhab (definition) (school of thought).