Based on the list "Who doesn't need a visa" from the Kosovo Ministry of Foreign Affairs, you do indeed need a visa as an Indian citizen.
Based on this QA website linked to by the Kosovo MOFA, it seems you can arrange to mail your passport to one of the visa issuing locations. Based on the "Visa application procedure" page, there is no requirement to show up in person to submit the visa request. Further, one of those locations is in Istanbul.
Additionally, based on the QA website linked, and the "Special Categories" section of the Kosovo MOFA, it seems holders of multi-entry Schengen visas do not need a special visa to enter Kosovo. However, it is unclear from the documentation if that visa is valid for the full 90 days, or for only 15 days (World Travel Guide says 15 days) .
If you take a look at a map of Kosovo, you will see that it only borders Serbia to the North. While not recognized by Serbia, Kosovo is recognized by over 100 nations and has permanent membership in several major global institutions. I suggest referring to this Travel.SE post about traveling between Serbia and Kosovo. However, if you plan to stay away from the northern part of Kosovo, there seems to be no reason why you would need a Serbian visa. The information in that post is echoed/backed-up by a blog post by the Belgian Liaison office in Pristina. Further, the UK government suggests that Serbian border guards might deny entry into Serbia if they see a stamp from Kosovo.
For transiting, as long as you do not leave the airport terminal, you should not have any need for a Turkish visa. This is common practice is most countries. Also, see this Travel.SE post regarding Indians needing transit visas through Turkey.
Based on the Turkish consulate government website, if you wish to visit Turkey as an Indian citizen you do indeed need a visa. Much like Kosovo, there is a special provision that will fast-track you for an e-Visa if you already have a Schengen visa.