Translation:It takes ten months.
The じっ pronunciation is the older form. At some point, people started using じゅっ under the influence of じゅう, basically replacing the /i/ vowel with /u/. Originally, 十 was pronounced zipu. By itself, it became zifu → ziu → zyū → jū, a series of regular sound changes (it was still spelled じふ in the pre-1946 orthography). However, when combined with counters such as ka, a different set of changes occurred, zipuka → zipka → zikka → jikka
It is. That's another way of pronouncing this kanji combination. In most counting in Japanese, whenじゅう is shortened to じゅっ it can also be shortened to じっ instead. Probably as a non-native Japanese speaker it's best to stick to じゅっ when saying it yourself, but be able to recognize what a native speaker is saying when they say じっ.
I imagine somebody got tired of writing the whole 箇 and just stopped after the first few strokes, which do look almost identical to a small ケ. And then it caught on. I wouldn't be surprised if there was a time it was still written in the upper left, where it would be if the rest of the 箇 was there.
I'm not sure what you mean; there is no katakana in this sentence. 「十ヶ月かかります」
You may just be seeing a different font used for the hiragana り than you are used to. Some fonts show the lift of the calligraphy brush around the curve while others show the slight drag between lines. Duo doesn't have a default font so it uses whatever the default is for the platform you are using. Switching browsers/browser updates may change the font displayed.
Katakana リ is much more angular/straight lined.