What does instagram have to do with Uni application? If you just use it as an easy to use online portfolio, there are better, free options. Google ‘free online portfolio builder.for photography’ etc. This way you can show off your photos big and less compressed.
IG can normally only be viewed on a phone and thats not ideal for showing off your photography to the university, they are much more likely using a laptop to access your links rather than a phone.
All the descriptions sounded pretty common especially if the photographer is starting out, building their portfolio. Everyone have to start somewhere and most starting cannot afford to pay models. (including taking a while to send photo back.)
There are things you can do to safe guard yourself: check their portfolio, use image-search on their photos to check if it is really theirs, Google their name, ask for the IG etc of other people they have photographed then contact those people to ask about their experience.
And try to insist on bringing a friend to observe the shoot. If the photographer refused, don’t go.
From my personal experience all the successful and efficient family photo sessions that involve toddlers, were never conducted by a photographer alone. There were always an assistant of some sorts who actively get the kids’ attention, cause them to smile or laugh, and potentially stop your Christmas tree being knocked down.