The objective of religion is Tazkiya (‘Purity‘) – both of soul and of body.
And this purity is required in every sphere of life. And since we have been given this keen sense of purity as a part of our nature, therefore most of this has been left to our natural inclination towards purity, especially the purity of body. The same has been emphasized on many occasions in religion and some basic form of purity (e.g. cleaning oneself after excretion and and urination, cleanign pubic hair, cutting nails, circumcision etc.) has been instituted as Sunnah by the Prophet (sws).
As far as other forms of impurity (e.g. urine and faeces of animals) are concerned, it is natural to feel disgust from them and one should try to clean them as soon as one may encounter such a situation. Our nature doesn’t differentiate between Halal animals and Haram animals when it comes to impuritues of urine and faeces. Therefore, the arguments presented by scholars who declare urines and faeces of halal animals to be pure are very weak and unsubstantiated.
As far as Salah is concerned, if one has fulfilled all the prerequisites of Salah, the Salah would be valid. The rest are matters of etiquettes (‘Adab‘) e.g. clean and decent clothing, clean place of prayer, women wearing a headcovering for Salah etc. which should be observed unless one is faced with a situation where one maynot be able to observe any of such etiquettes, in such a situation, leaving of any such etiquette won’t render one’s Salah ineffective. The Hadith which is widely used to support traditional view where Prophet (sws) allowed some men to pray in Sheep Pen or Camel Pen had to do with the same argument i.e. because of nature of one’s job, one was unable to achieve that ‘etiquette level of purity‘, hence they were allowed to pray in that place. It was not an absolute permission and neither did it have anything to do with Halal animals or Haram animals.