Does performing the intention of permissibility (al-ihlāl) after the talbiyah, or some hair falling out when combing it, or a piece of a nail breaking, or not adhering to all the prescribed supplications during the sa'y, or trimming only the ends of the hair instead of shortening it all over, invalidate the 'umrah and necessitate an expiation?

Your 'Umrah is valid, God willing. Stipulating conditions in Hajj or 'Umrah is not obligatory, so omitting it does not invalidate the 'Umrah. It is not advisable for a pilgrim in a state of ihram to comb his hair. If hairs fall out due to combing, it is more cautious to give charity. There is no issue with a nail that falls off by itself, but it is more cautious to give a mudd of food as charity. Reciting the ma'thur supplications at Safa and Marwah or during the sa'i is sunnah, not obligatory. The sufficient amount for a woman to shorten her hair when exiting ihram is a matter of differing opinions, and what you did is sufficient according to some scholars. Doubt about the validity of the 'Umrah after the worship has been performed does not affect it and has no consequences.