Intended Solution:
Prof. asks Katy to give her parents phone number. Katy's reply was an unusual number. But if we think of a dialpad, typing numbers also create alphabets.
The number was $T_8$ $B_{22}$ $H_{44}$ $I_{444}$ $D_3$ $K_{55}$, short form for "To Be Honest I Don't Know".
Prof. says she's lying and the order is different. Till now there's no clue about what order is.
Notice that Katy's next response has 6 words to be exact, which hint towards order of the letters $T, B, H, I, D, K$
Prof. then says something about lengths. Now if you observe the lengths of words from Katy's previous response, they are $6, 1, 4, 2, 3, 5$
This suggests an ordering of previous ordered letters we obtained. The new order would therefore be $KTI.BHD$
Next sentence has the phrase "dashes and dots" referring to Morse Code.
If we convert the above letters into Morse, including the dot additionally introduced by Katy, we have:
$ -.- / - / .. /./ -... / .... / -.. $
If we check the whole conversation, Katy says "I flipped", "to the point", and "get connected". Hence we try to flip dots into dashes, and dashes into dots for all alphabets before the full stop (i.e. $K T I$ ) and connect them to get the string:
$ .-. . -- . -... .... -.. $
If we program a code, you might find there are 73444 possible strings which would satisfy above Morse Code when concatenated (I can assure you that only 1 common girl's name present in all these possible names). Some of these names start with L, and do not follow by a vowel, which get directly eliminated. Most of the other names get eliminated when checked by hand. But to make it shorter, if we guess length = 6, assuming 6s to be dominant in the question, then we only have 306 possibilities, and by hovering over you might find that Katy's Aunt's Name is actually nothing but:
$ R A C H E L $ , (morse code is $ .-. /.- /-.-. /.... /. /.-..$)
Loose hints:
If we check the number of special characters in each dialogue, they correspond to the length of characters to partition the concatenated morse string
©law, true love, professor - is a F.R.I.E.N.D.S. reference to lobsters representing Ross (Prof) and Rachel (Katy's Aunt)
Author's comments:
I remember spending only 1-2 hrs creating this question, and never got it cross-checked. If you find any other answers, feel free to post them here. I hope you enjoyed the solution!