The definite article is required to differentiate the city from another word with the same spelling جرطوم (“trunk of an elephant”).
In the rare case that you are talking about the elephant trunk and the city of Khartoum at the same time, one could use «المدينة الخرطوم» to specify the city.
You're absolutely correct! 3aaSimatu because of iDaafa الإضافة (if we follow the Standard Grammar).
The audio says 3aaSimati because Duolingo try so hard to eliminate the ending sounds by making them in sukuun (dead) and neglecting the Standard Grammar. They say this way is less difficult for learning Arabic.
the dead ة ta marbuTa meet ال, they thought that it is like ت ta in sukuun meet ال and they read it as -til (or -tis in this case).
There is another way to pronounce ta marbuTa in sukun without destroying the Grammar, like: 3aaSima as-sudaan (or 'aashimah as-sudaan if we follow another Arabic transliteration method).
It's because when ta marbuTa is in sukun, we spell it as ه ha in sukun (h).
The complete endings will be:
الخرطومُ عصيمةُ السودانِ.
If your question means "why the capital is in ma3rifa معرفة (with ال, ie. we know exactly which the capital it is)", it is because of the 2iDaafa (possessive form) in "3aaSimatus-sudaan" عاصمةُ السودانِ literally means "Sudan's capital". (So, because of Sudan is definite, then the word "capital" is also definite).
If you mean why the capital is (in English Grammar), it will have a different explanation because ال and the are not totally in the same principle. And, I suppose it would be at the advanced level a bit.