All of your questions have been answered many times on pprune before. I've tried to help you anyway.
For Single Engine: GFT Day and Night are to be done with the instructor (48min each as mentioned in this thread before).
But the flowchart also states that I need 3 "solo" take offs and landings.
So, am I required to do the solo TO/LNDGs separately and get a certificate from my instructor for that?
As far as I understand things, you're supposed to log your GFT as In Command Under Supervision or ICUS. Not sure how it is in the US but in I'm allowed to do that in Australia.
The 3 solo take offs and landings are supposed to be a part of the GFT. All in one flight. By asking for 3 "solo" take offs and landings DGCA are actually saying that you have to be in command of the aircraft for a minimum of 3 take offs and landings. If you log your GFT as ICUS, it means that you were in command of the aircraft throughout your GFT flight, while being supervised by the instructor. Which is obviously how a test works.
Hope that makes sense.
Can I log some instrument time with my instructor during the GFTs? (as part of the 5 hours instrument time required for Instrument endorsement on DGCA CPL)
Not 100% sure on this one. But not a sensible thing to do. GFTs aren't long enough for you to have time to long instrument time. All you do is circuits and training area stuff. Where is the space for instrument flying?
I believe that the 120NM and 250NM cross country "tests" are to be performed with the instructor, am I right?
Tests obviously have to be flown with an instructor.
One more question, can I log some instrument time during the cross country tests?
This is much better idea compared to logging instrument time on your GFTs.