Is it obligatory to cleanse oneself with the left hand only? And does prayer become invalid when using the right hand due to the difficulty of cleansing with the left and the feeling of incomplete purification?

The recommendation to perform istinja' (ablution after defecation or urination) with the left hand is the opinion of the majority of jurists. It is disliked to touch the penis with the right hand during istinja', based on the hadith of Abu Qatadah: "None of you should touch his penis with his right hand while he is urinating, nor should he clean himself from the privy with his right hand." The majority interpreted this prohibition as a recommendation (tanzih), while the Zahirites interpreted it as a strict prohibition (tahrim). Prayer is not invalidated by performing istinja' with the right hand. The prohibition applies to direct contact of the hand with the penis or impurities; however, there is no harm in using the hand to pour water or wipe.