Yes, one can have any countable ordering. Indeed any countable totally ordered set can be embedded in $\mathbb{Q}$. Write your ordered set as ${a_1,a_2,\ldots}$ and define embedding redursively: there is always an interval to slot $a_n$ into.

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.