The Assumption Of Purity, Not Filth

In fiqh of taharah, we are told to assume that the floor of a non-Muslim's washroom (defined as one owned and/or frequently used by non-Muslims) is najis.  is this correct?

Walaikum Assalam,

Nope. The default assumption about things, is that they are pure (tahir), unless it is certainly established otherwise. This includes the washroom floors of non-Muslims, and those in frequently-used places. Unless any trace (=smell, colour, or taste) or actual body of filth (najasa) is perceptible, we assume it is pure. It is considered lowly, though, and should be avoided because of this. But it is not filthy. [cf: Ibn Abidin's Hashiya, Bab al-Tahara, and the section on waswasa in Imam al-Barkawi's al-Tariqa al-Muhammadiyya]

In fact, Shaykh Umar al-Shiyyab told me that his spiritual guide, Shaykh Muhammad Sa`id al-Kurdi, ordered a brother with a lot of misgivings (waswasa) about issues of tahara to sit on the floor of the toilet in his mosque for 30 minutes, and then to pray without washing his clothes, to get rid of these misgivings. Radical? Yes. But it works. Similar approaches to defusing waswasa were mentioned by others. It comes down, as Imam al-Barkawi and others explain, to ignoring the misgivings, and operating only on the apparent certainty. One does not do "if-then" analysis analysis with najasa. ("Well, IF some urine had fallen on the ground, THEN... it must be najis.)

what if it has been raining and the sidewalk is wet and one is walking on the sidewalk, but someone happens to be walking their dog a few paces ahead of one? would one's shoes be considered unclean since they come in contact with a wet sidewalk that a dog has touched a few paces ahead?

The only thing najis about dogs is their saliva and meat. [as mentioned by Ibn Abidin and others]

Wassalam,
Faraz