Relation between REG and NLOGTIME? What is known about the relation between the class of regular lanuages and NLOGTIME? Is any class contained in another one?

You have some choice of how to define NLOGTIME to get one or both inlcusions; cf., e.g., Regan &amp; Vollmer, Gap-languages and log-time complexity classes, TCS 188 (1997), pp. 101–116.
Parity is a regular language not computable in NLOGTIME, or even in nonuniform AC^0. The set of all strings whose length is a power of 2 is a non-regular language computable in DLOGTIME.