My final destination is LED. Sure i may try to re-book on another carriers in IST, but if they agree to do it in IST they could as well re-book me on QR from DOH (i mean, if they still have to buy me a biz class ticket on another airline...)
I have neither confirmed nor rejected the changes. But my understanding is that they automatically re-scheduled me on TK783, departing at 5.55am from DOH. Why they haven't automatically re-scheduled the second flight is a mystery to me.
I am in BKK now and will contact their sales office here (they dont answer the phone, so will have to go there). I am only connecting in DOH so haven't planned to leave the airport there.

Yes, there is a good chance that the DOH airport staff will be prepared to rebook you to QR once they realise that you will miss the IST-LED flight that you are currently booked on. But only on the day of departure. There is close to zero chance that TK will rebook you like this in advance. This also means to that going to the BKK office will most likely be fruitless. Actually, your best chance of getting rebooked to QR from DOH or another carrier from IST is by leaving the reservation as it is and let the staff at DOH airport deal with it on the day of departure - or if they refuse insisting that you need to travel to IST on TK 783, at IST transfer desks. Once you contact TK and someone rebooks the IST-LED flight it will be much more difficult to get DOH staff to intervene as it will no longer appear to be a problem that they need to deal with. Letting you travel as booked to IST on TK 783 will be far easier for them.

It is quite common that rebooking a cancelled flight doesn't lead to rebooking of any subsequent connecting flight. Neither the first nor the last time this has happens. Subsequent flights will only be rebooked when a TK agent looks into your booking and realise that there is a problem.

Actually, if your incoming flight is early or even on time, there is a chance that you will be able to make that connection at IST (if the DOH check-in agents let you travel on this itinerary; doubtful as you are below MCT at IST, i.e. the DOH agents may try to insist that your IST-LED flight is rebooked).

So here's how the story ended.
The agent in the sales office in Bangkok called the sales office agent in Doha, who re-booked me on DOH-DME (D class, haha). He said he couldn't book me on DOH-DME-LED (the last leg being on S7) as it is a mix of two airlines. Not sure why is that, but i agreed and just purchased the DME-LED ticket myself. Hope TK is gong to reimburse me.
The other option offered to me was DOH-LED direct, on the next day. But that would require an overnight in Doha, and they did not confirm they were about to pay for my hotel.
On the one hand, once i was in the Bangkok office, the process was relatively quick and easy (took about one hour).
On the other hand, i have lost the entire day trying to sort it out, calling different call centers, getting to and from their city office.