Does touching a dog make my wudu invalid?

No — merely touching a dog does not break wudu in any of the four Sunni madhabs; wudu is only nullified by the specific things listed in Q1 (discharge from the private parts, sleep, loss of consciousness, and the disputed items like bleeding or opposite-gender touch), and touching an animal simply isn't on that list. This is a common misconception — people often confuse "my wudu broke" with a separate and different issue: whether the dog's saliva made a part of your body or clothing physically impure (najasah), which is a matter for cleanliness before prayer, not for wudu itself.

On that separate impurity question, there is a genuine madhab difference worth knowing: the majority (Hanafi, Shafi'i, and Hanbali schools) consider a dog's saliva/wetness impure and requiring washing if it touches skin or clothing (based on the hadith about washing a vessel a dog drank from seven times, once with earth) — though the Hanafi school confines that specific seven-times rule mainly to the vessel-licking scenario and treats ordinary wet contact with a dog more leniently, requiring regular washing rather than the full seven-times ritual. The Maliki school takes the most lenient position of all, holding that dogs are not inherently impure animals at all, and that the vessel-washing hadith is understood as a matter of hygiene/devotional practice rather than proof of najasah — so touching a dry-coated dog, in the Maliki view, doesn't require any washing at all.

Source: hadith on washing a vessel after a dog drinks from it, Sahih Muslim; general fiqh differences on canine najasah across the Hanafi, Shafi'i, Hanbali, and Maliki schools.