Is it obligatory to wash the bedding again after it has been stepped on by two cats known to urinate everywhere?

It is clear that you are experiencing waswas (obsessive whispering/doubts). You must disregard them and pay no attention to them. The clothes and bodies of children are presumed to be pure until their impurity is proven with certainty. Therefore, you should not judge the clothes of your neighbor's children to be impure merely because they played with a dog. Your prayer is valid when you are near them, and this is not an excuse to miss congregational prayer; rather, it is from the waswas of Satan. Increase your supplications and seek refuge with Allah from Satan, and strive to attend congregational prayers. As for the second question, it should be submitted independently.