> am intrigued by the idea that I may be looking in the wrong place.
No, not necessarily. You're right, Pliner and the German versions Plainer or also Pleiner (in German the same phonetically) do not sound *Czech*. They definitely sound like German (there may still be a Jewish form as well). The Austrian phone directory has some 90 entries for Pleiner and 158 Plainer - meaning it's not really rare. You'll probably find hundreds of them in Germany as well.
As to seeming "not Bohemian" - before the (more than unfortunate) invention of European Nationalism Bohemians were both - Czech and German, simply people living in Bohemia. The name Plainer would've been perfectly 'Bohemian'.
> Why would they have 3 children before marrying? Did they flee Bohemia? Was she Bohemian and he German and no one wanted them to be together? <
I don't know the answer. Just keep in mind that in several areas in Europe (maybe elsewhere too) a high percentage of births were (often still are) illegitimate, i.e. before marrying and even without marrying ever. I know of regions where these were some 40% of all births.