Cinque Terre Card (faq)
No, there is no entrance fee to the Cinque Terre National Park.
Yes, dogs are allowed with leash and muzzle...
On all regional trains – even those not marked with the special symbol – passengers are permitted to travel with a fully closed folding bicycle (even if not in the special bag) free of charge, provided that the size does not exceed 110 x 80 x 45 cm and does not cause danger or inconvenience to other passengers.
No, you don't.
Whether you need a Cinque Terre Train Card or not, depends on how many journeys by train between the villages you intend to make. The card costs €18,20 a day for an adult. A single train ticket costs €5 per journey if bought point-to-point.