My sense is that because disinda (I can't do the letters properly on my iPad) is a locative, it implies the place where an action occurs, and not the direction of the action. I could be wrong but I don't think your translation would be accurate. Your sentence would read 'Istanbuldan kosuyoruz'.
"Outside Istanbul" is acceptable, even though Duolingo considers it a typo.
Actually, "outside of" is more common in American English than in British English, so "outside" alone is perfectly fine.
Not sure, but I believe the Turkish here is making a distinction that cannot be easily made in English, and the answer key tries to (artificially) make this distinction by refusing the translation out of Istanbul and only accepting the (basically synonymous) outside of Istanbul. The locative case (dışında) names the place wherein you are running (that is, from one point outside of Istanbul to another point outside of Istanbul). The dative case (dışarı) shows movement toward a goal (that is, where you are running to, from one point INSIDE of Istanbul to another point OUTSIDE of Istanbul). Both of these meanings in English could be expressed with either I am running out / outside of Istanbul. Hence the confusion. English is not a translation of Turkish, nor is Turkish a translation of English. Learning language purely through translations always has its difficulties. :-)
Bilmek is one the verbs which very often are used with present continuous tense in Turkish.
The following most common verbs that aren't normally used in the simple present tense (in English) are mostly used in the present continuous tense in Turkish: istemek ( to want), duymak, işitmek (to hear), görmek (to see), kokmak (to smell [to have a particular smell]), sevmek (to love / to like), nefret etmek (to hate), inanmak (to believe), anlamak (to understand), bilmek (to know [to have information about something]), hatırlamak (to remember), oturmak/yaşamak (to live [in a place]), tanımak (to know [to be familiar with a person]), unutmak (to forget).
I hope this helps :)
An English verb can usually be classified as either an "action verb" or a "stative verb". As the name implies stative verbs don't normally describe an ACTION taking place ('run', 'speak') but rather a STATE ('know', 'like'). Stative verbs are not used in the continuous form in English.