Are there integral domains which admit ordinalvalued Euclidean functions but not $\mathbb{N}$valued Euclidean functions?

Yes, they exist. Even if the problem is left open in the the papers of T. Motzkin and P. Samuel cited in Comparing different Euclidean algorithms on a Euclidean domain the problem is solved in J. Hiblot, Des Anneaux euclidiens dont le plus petit algorithms n'est pas valeurs finies, C. R. Acad. Sci. Paris, 281 (1975), 411414 with correction in vol 287. M. Nagata, On Euclid algorithm. C. P. Ramanujama tribute, pp. 175 Tata Inst. Fund. Res. Studies in Math., 8, Springer, BerlinNew York, 1978. See also: Pag. 195 in P. M. Cohn, Free Ideal Rings and Localization in General Rings Rod Downey, Euclidean Domains and Euclidean Functions, http://homepages.mcs.vuw.ac.nz/~downey/ntu_13.pdf Pete L. Clark, A note on euclidean order types, http://arxiv.org/abs/1208.0977 

