You ask that all aspects of the question presented by the OP be addressed. So first, to restate the specific points enumerated from the OP:
- is Hashem indeed speaking L'H, and
- is this only permissible because he is God, and
- is this a case we shouldn't try and emulate Hashem?
There is general teaching found within both the written and oral Torah that G-d keeps all His commandments. This is based upon Tehillim 147:19 which says:
He issued His commands to Jacob, His statutes and rules to Israel.
This same theme is also in Brachot 6a in the name of Rabbi Yitzchok who asks where is it written that G-d wears tefillin.
And so, just as we have a commandment prohibiting speaking Lashon HaRa, so too does G-d. But while some of us may stumble in this mitzvah, G-d does not.
And this answers points 1 and 2 from the OP. No, G-d is not speaking Lashon HaRa because He keeps His commandments and because that first premise is incorrect, point 2 also falls away.
The 3rd point in the question is more general. The OP postulates from point 2, if G-d is speaking Lashon HaRa against Sarah in the Parsha quoted and then follows to ask in point 3 is this an example where we shouldn't follow G-d's behavior?
In truth, a more careful reading of the Torah text is needed. The Torah discussing G-d's interaction with Avraham and Sarah in Bereshit 18:9-15 is actually via His angels, not G-d directly. So strictly speaking, this story is really asking should we emulate this type of behavior in regard to dealing with His Angels.
But if we deal with what the OP actually asked, the general Torah teaching is from Mishlei 3:6
In all your ways know (meaning connect, as in cleave with) Him, And He will make your paths smooth.
which teaches that we should emulate G-d's behavior and teaching in all areas, like is expressed by Metzudat David to this posuk in Mishlei.
And if the side issue from this question is considered, namely should we behave with G-d's Angels in the way that is taught here in the Torah?
The response would be yes. But what the specific teaching is here, as it relates to interactions with Angels, is a more complex subject. This would fall into the area discussed in Malbim's commentary to the same posuk in Mishlei which distinguishes between the broad general ways of G-d and the more specific paths of wisdom.