What is a drivers’s responsibility to animals ridden, led, or herded on the road?
Drivers must yield to animal riders and livestock (horses, mules, donkeys, cattle, swine, sheep or goats).  Animal riders and herders have the same roadway rights and responsibilities as motorists, except where the laws have no application.  Approach animals cautiously.  Pass wide and slow.  Local communities can regulate animals such that they’re restricted to certain paths or prohibited on certain streets.   ORS 814.140, 810.100

If the animal rider or livestock herder raises their hand, that’s a distress signal.  Motorists must stop and, if requested, turn off their engines until the livestock is under control.  Failure to comply is a class B traffic violation. ORS 811.510, 814.150

What are city-level restrictions regarding animal-related transportation?
Portland defines “traffic” as including “ridden or herded animals”. Portland City Code 16.90.365

1.) Equine cannot be ridden or led in any park except upon a roadway, designated parking area, or designated bridle path. Corvallis Municipal Code 5.03.050.080.01
2.) No livestock may be led, driven, or ridden on any sidewalk.  Corvallis Municipal Code 5.03.050.020.06(7)(b)

What is a driver’s responsibility if they hit a domestic animal?
Stop.  Attend to the animal.  Contact the owners, or if unavailable, the police.  Failure to do these is a class B traffic violation.  ORS 811.710

Are there extra legal penalties if I hit someone riding an animal?  
Animal riders are “vulnerable road users”, which is a protected group. If you seriously injure a vulnerable road user while “careless driving”, additional penalties can include multiple court appearances, requirement to take a traffic safety course, 100-200 hours of community service related to driver improvement and providing public education on traffic safety, $12,500 fine, and 1 year driver license suspension. ORS 801.608, ORS_811.135, Explanation from a traffic layer