Partial order, total order, and version order in transaction histories
for what it concerns Question 1:

there is a theoretical result from 1930 [1] that states that 

any partial order $S$ can be extended to a total order $S'$ which contains $S$. 

This result is known as the "Order Extension Principle"; the proof of this result 
uses the Axiom of Choice (I am not aware whether there is an alternative proof that does not use it). The paper is written in French, but you can find a proof at 
https://www.proofwiki.org/wiki/Order-Extension_Principle. 

When Adya states that he chooses the real-time order of events, the best guess 
is that he assumes an implementation of the database; in this case, every history $H$ 
corresponds to an execution of the database, where events are totally ordered; such an order is the real-time order.
More specifically, by requiring that the partial order $&lt;$ in a history $H$ is extended by the real-time (total) order, Adya imposes that whenever $e_1  &lt; e_2$ in $H$, then the instant of time $t_1$ at which the event $e_1$ takes place (in the execution of the database that leads to $H$) is less than the instant of time $t_2$ at which $e_2$ takes place.
Choosing the real-time order of events to extend the partial order of a history $H$ is needed if one wants to prove the correctness of an abstract specification (i.e. set of properties that a history has to satisfy) with respect to a given specification.

Let's turn to Question 2: 

in this case it seems to me that Adya wants to stress the fact that he allows the implementation of a database to access a version of an object which precedes (in the version order) the latest version installed. In practice, choosing an earlier version of an object could be either the result of the database not being able to access such a version (e.g. the lost update anomaly in causal consistency), or the user explicitly requesting to access an older version of an object (e.g. accessing an older version of a revision in a SVN repository).

Hope this helps, 
Andrea Cerone.

[1] Edward Szpilrajn - sur l'extension de l'ordre partiel