Ms. Smith "having been instructed in the rudiments of jurisprudence duly satisfied the Order [of examining professors as to her learning] when examined in the customary fashion".

'Propaedeutica' is Greek and means 'prelimimary education', here probably with reference to what's called 'pre-law' in the USA.

'Institutione interrogata' is an ablative absolute for 'cum institutione interrogata esset', 'when/since she, etc.'. 'Institutione' is an ablative of means ('institutio' here = 'established custom').