You are right about Prince Arthur. He was next in line after the death of Richard I but the throne was seized by John. John killed Arthur so the next in line was his own son. The only other example is when Henry IV took the throne when the heir presumptive was Edmund Mortimer, Earl of March.
Correct, the first of every name. Actually regnal numbers did not come in use till the reign of Edward III. Before that kings were distinguished by name Fitz (son of) father's name. The confusion arose with the accession of Edward III as that meant there were now two Edward Fitz Edwards.