1) I have recently learned that when toilets are flushed, small water droplets containing poop particles and other human waste (e.g urine) splash from the toilet bowl and come underneath the toilet lid and on top of the toilet lid, as well as perhaps some other areas (e.g landing 3 feet in front of you). For this reason, it is advised to close the toilet lid before flushing. From the Islamic point of view, does the toilet lid need to be washed when these water droplets containing human waste land on the toilet lid?

2) Similarily, I have read that microscopic particles containing human waste may land on the toilet lid, near the toilet, on the floor, or as far as 3-6 feet away from the toilet in a phenomenon known as the "plume effect." Would cleaning the areas (e.g. toilet lid, floor, other parts of toilet) be necessary in this case as well?

1) Yes, it would need to be purified if impurity gets on it
2) It would be necessary if you know for sure it has spread there via the droplets. However, if you cannot see it because it is microscopic then you can assume it is not there and thus doesn’t require purifying. Establishing that impurity is there has to be by normal means (seeing with the eye and being certain of it)