In the public domain, there are nine frogs and one sheretz. A person touches one but doesn't know which. Is he impure?
Halacha 2: public-domain doubt is always pure — even when nine are sheratzim and one is a frog, if the contact occurred in the public domain, the person is pure unless he says he is certainly impure.
Question 2
A person lost a vessel during the day and found it in the public domain later that SAME day. Is it impure?
Halacha 7 (halacha 6): a vessel lost by day and found the same day in the public domain is pure — but if found the next day, or if the night passed over it, it is presumed impure.
Question 3
A gravely ill person is in a private domain, loses consciousness, then is moved to a public domain. While in the public domain, is his uncertain status creating impurity?
Halacha 14 (halacha 13): the Sages explicitly ruled on this — when the person is in the public domain, his doubtful impurity is pure; when brought back to private domain, it is impure again.
Question 4
An impure person stood over a cistern of wine and a saliva drop may have fallen in. Is the wine impure?
Halacha 15 (last halacha): a wine cistern's rim catches an incoming drop of saliva — therefore the doubt is resolved as pure in all places. An oil cistern (which lacks this catching mechanism) in a private domain would be impure.
Question 5
A person spread out vessels in a private domain, stepped away to retrieve them after they fell, and found them. Are they impure?
Halacha 9 (halacha 8): vessels spread in private domain are impure unless supervised. If they fell and he walked to get them, they are impure because they went out of his sight — unlike depositing or forgetting which is less common.