The first one should be two sentences since there's a statement and a question. I'm not sure "earnest" is the best adjective for describing a design. It's not clear what you mean to say. In the first one, "however" is an interrogative adverb. In the second one, however is a conjunction. "However" is a conjunction of contrast, but I'm not sure it's appropriate there since it's unclear what your contrasting. By the way, the question would sound more natural with a different word order: How did they ever manage to paint such fine detail?

The first one can be one sentence with a completely different meaning in a more old fashioned sense too, if you split 'however' into 'how' and 'ever'. 'The design of that china plate is beautiful - how ever did they manage to paint such fine detail?' This sentence now means: 'The design of the china plate is beautiful - it seems amazing to me that they could manage to paint such fine and intricate details on it!' Jonathan is correct (below) to point out that, here, the use of 'earnest' (meaning honest, or sincere) is not an accurate adjective to use.

The first sentence praises the artist who created the plate. "However did they manage...?" = "How in the world did they manage...?" = "How did they ever manage...?" = "How on earth did they manage...?" The second sentence doesn't make a lot of sense. When you say "however", you imply that there is some sort of contrast. However(!), I don't see what the contrast would be between "earnest" and "fine detail".

Earnest would be more appropriate to describe the effort that was put into the design and production process, rather than the design itself. You may talk about an intricate design instead. ;however did they manage to paint such fine detail? - This would be a rhetorical question to convey how amazed the person is by the finished product. Using a semi-colon would be a good way to tag the question onto the statement. "however, they managed to paint such fine detail." - This would indicate a contradiction, which wouldn't work in this case as there doesn't appear to be a contradiction. Note that I added a comma after however. This is because this word is followed by a pause when spoken.

Like Jonathan suggested, there are better adjectives to use for describing a design.