The incorporation of Phosphorus into (Ga,Mn)As epilayers allows for the tuning of the magnetic easy axis from in-plane to perpendicular-to-plane without the need for a (Ga,In)As template. For perpendicular easy axis, using magneto-optical imaging a self-organized pattern of up- and down-magnetized domains is observed for the first time in a diluted magnetic semiconductor. Combining Kerr microscopy, magnetometry and ferromagnetic resonance spectroscopy, the exchange constant and the domain wall width parameter are obtained as a function of temperature. The former quantifies the effective Mn-Mn ferromagnetic interaction. The latter is a key parameter for domain wall dynamics. The comparison with results obtained for (Ga,Mn)As/(Ga,In)As reveals the improved quality of the (Ga,Mn)As$_{1-y}$P$_y$ layers regarding domain wall pinning, an increase of the domain wall width parameter and of the effective Mn-Mn spin coupling. However, at constant Mn doping, no significant increase of this coupling is found with increasing P concentration in the investigated range.