Can a body $X$ apply a gravitational force on a body $Y$ without itself being acted on by a gravitational force from $Y$? Just per intuition, we deduce from Newton's third law that if $A$ applies force on $B$ then $B$ applies force on $A$. But in the case of gravitation, they both apply force on each other. So which force came first as an action, and which came later as a reaction?
Both forces appear at the same time.
No, a force cannot be applied by one body on another body without the first body feeling a reaction force. This is due to Newton's third law, as you mention.
For the action/reaction force pair we can't say that one "came first". Both forces of the pair must be present simultaneously. Calling one the action and the other the reaction is only done as an aid for the intuition; it has no physical significance. They both appear exactly simultaneously and one did not "come first".
Neither comes first, they are mutual. In the Newtonian model gravity acts instantaneously at-a-distance (unlike Relativity, where gravity propagates at the speed of light).
Which force you consider the action depends on which one grabs your attention first. The other one then has to be the reaction. But the distinction in this case has no physical meaning (unlike the case where some agent initiates an action, such as when someone steps off a dinghy).