Sorry for the delayed response - just got in.
28a is a bit funny, as it's not what one might expect from a prestigious newspaper. I wonder if the editor knows?
18a: I put PERTS as, I think, others discussed before, as STREP backwards seems compelling. The way I see it is ... PER (latin 'through' - could have meaning 'hardy' as in 'we will get through this' ?) and TS for types. I wonder what the compiler actually had in mind though; would be interesting to know.
26d led me all around the block. At first I had BREDA (ie braid.. maybe, if concerning hair: dreadlock). Later, I had it as DREDA (diminutive of Eldreda, feminine form of Eldred/Aldred, meaning 'old counsel'). Finally, realising the final 'A' messed up YARD, I had a third shot at it to get the (I hope) correct answer.
Brain hurting! .. Jim