Would an undignified planet in the 10th cause problems to the 4th house by 180-degree aspect?
Yes, but make sure it truly is aspecting the 4th house. Do that by calculating the aspect to the “cusp” of the 4th, which in a whole sign house system is the ascendant’s degree transposed to the 4th house.
So, if your ascendant is 10º of Pisces, then Gemini creates your 4th house, but 10º Gemini is the most sensitive point, the “cusp”, the “door to the house.” I planet at 10º Sagittarius will very strongly influence the 4th house, but a planet at 1º Sagittarius will not – because of how degree-based aspects work.
The 10th seems a good place for most planets, but I assume the dignity is a major factor. Could an undignified planet be more problematic because of its prominence by being situated here?
Yes, the 9th, 10th, and 11th houses – being highest in the sky, are good for both malefics and benefics. A slight exception is the Moon in the 10th, which presents some clash between inwardness (Moon) and outwardness (10th). But if any planet is undignified in the 10th it is problematic, because being in the 10th gives it a spotlight.
Take this a few steps further. Calculate the actual prominence of each planet. To have undignified planets in prominent conditions is undesirable. To have dignified planets in prominent positions is most desirable.
How does the moon cope in the 10th? (To me it seems a little bit out of its comfort zone.)
Yes. It signifies a clash between a wanting to be important in the world, but being markedly and prominently shy – which interferes with fulfilling that desire to be prominent in the world.
Of course a dignified Moon will downplay the negativity mentioned above, and an undignified Moon will magnify it.
Vic DiCara (vicdicara.com)