lo.logic - Order types of positive reals

Yes, one can have any countable ordering. Indeed any countable totally
ordered set can be embedded in $mathbb{Q}$. Write your ordered set as
$ lbrace a_1,a_2,ldots rbrace $
and define the embedding recursively: once you have placed $a_1,ldots,a_{n-1}$
there will always be an interval to slot $a_n$ into.