No, it’s not necessary to bake the normal map. The normal map is generated from the height map on the fly. The default height level is 0.5. So, you need adjust the brush height level, all above 0.5 is positive, anything below 0.5 is negative. After you painted in the height map, the normal map gets automatically generated.
So, to paint only the height, you can disable each other channel individually by clicking on the preview image in the bottom. A "green" border means it is activated.
The list in the left bottom corner is only to select the current shown channel in the 2D View.
Normally it should export your normal map. Can you send me or post a screenshot of your export settings and your height map in the software? Maybe it’s a bug that your normal map doesn’t get exported. Please post it here or send me an email at support@unionbytes.de.