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I have just been there.
What options do we have? Can we go there by bus / foot? Can we organize a guided tour?
There are no issue crossing the "border" between Kosovo and northern Kosovo. In fact, there is no "border". There are some KFOR police, but in fact much smaller presence, than in some regions (e.g. Prizren). There are absolutely no issues crossing it by foot, no documents are checked. Organized tours are definitely possible, but one has to find a guide.
How safe is it today?
It feels safe, it looks safe. We have not seen any Kosovo / Serbia related issues. There are tons of Serbian flags, attributes, etc once you cross the bridge in Mitrovica. However, one has to be aware about incidents like this which probably explains why authorities do not recommend going to Northern Kosovo.
Can we potentially have troubles crossing the βbridge borderβ in Mitrovica?
As mentioned above there are absolutely no issues crossing the bridge.