[AllylicAzideUMN]

Every organic chemistry student knows that a molecule with an internal mirror plane is meso and therefore identical to its enantiomer. However, what if the stereocenter in question lies on the axis of this mirror plane, like in the case of the methyl decalins depicted below? My intuition is that 1 and 2 are diastereomers of each other (because the only internal mirror plane includes the only stereocenters) and that only 3 (inversion center) and 4 (internal mirror plane) are meso of the four. Is my thinking correct?

[phth]

Not quite. 1,2 are enantiomers because the lower stereo center has 2 identical groups attached. 3,4 are enantiomers.  step 1) Assign R,S config.  step 2) Did all of the symbols flip? Yes:enantiomers No: diastereomers.  For many, it's easier to think in terms of R's and S's than in 3 dimensions.

[clarkstill]

None of these compounds is chiral, hence none are enantiomers of one another (since all possess a mirror plane down the middle).

1 and 2 are diastereomers of each other, and 3 and 4 are diastereomers of each other. None of the compounds is meso, since for a stereoisomer to be meso it must be "of a set of diastereoisomers which also includes one or more chiral members" (https://goldbook.iupac.org/html/M/M03839.html)- there are no stereoisomers possible for any of these compounds that would be chiral since all possible stereoisomers would retain the mirror plane.

[rolnor]

They are  cis,trans-isomereres?

[Babcock_Hall]

Every organic chemistry student knows that a molecule with an internal mirror plane is meso and therefore identical to its enantiomer.

An achiral molecule may have internal symmetry, but it is not necessarily a meso-form.  An achiral molecule is identical to its mirror image, but a chiral molecule cannot be identical to its enantiomer.