Would this sentence be just as correct were one to leave out each "mí"?
I think so, although here Lúnasa and Iúil are genitive (since they each modify mí) and need to become dative when following i (so Lúnasa and Iúil, basically the same forms).
On the other hand, if it had been i mí Mheithimh (there is lenition because it modifies mí, which is a singular feminine noun), where Mheithimh is genitive, it would have become Meitheamh, Meitheamh being dative.
EDIT: it should be i mí an Mheithimh, because Meitheamh means "midsummer".
The ú in úll is broad while the second i in iúil is slender. So the L in iúil will be pronounced with sort of an English "y" ( like yuh, but take out the uh part) attached behind it. (According to my understanding of broad and slender consonants)
Not just you. In the font Duolingo uses capital <i> and lowercase <l> look the same.
There’s a curve at the bottom of the lower case L in Museo Sans Rounded (the Duolingo font), but it’s subtle and easily missed in small type sizes. (Its upper case i has no curve.)