I still have my visa for 30 days with multiple entry with maximum duration of 20 days.
I will presume from all you said this visa was issued by France.
Can I use that to travel on my own personally to Paris and Rome only, and go back to India from Rome?
Yes, not a problem, visas are issued for duration, not a fixed travel itin, the itin is necessary as one proof among many to prove you are a genuine visitor and not an illegal immigrant.
France is still my main destination. My plan is to now enter from Paris and stay for 4 days there and go to Rome and stay there for 3 days.
So far, still good. Your main destination is only relevant to see you are not trying to game the system by trying to stay, say, in Spain by using a visa issued by the French authorities. And you are not.
I can definitely carry the Thomas Cook documents regarding the cancellation
That's not needed, itin changes happen all the time.
although in the rare instance if they may ask me, can I say I had a health issue and had to reschedule the dates?
ABSOLUTELY NOT. Lying to immigration is the WORST IDEA possible. They are trained to find out, you are not trained to lie. You will get caught and you will get into trouble.
Will the visa application which had my original travel dates be available to them?
Of course, and it's almost irrelevant. Originally you filed from 17th to 27th and now you got a visa from 15th August to 15th September and now you want use the flexibility given to you and travel from 26th to 3rd. Your visa is valid, your intent to enter as a tourist matches your visa: go ahead. If asked why did you change, just tell the truth, even if it's as banal as "this fits my schedule better". The very reason they issued a slightly broader visa so if you change your stay by a week or two you don't need to go back and waste their time to get a new visa issued.
Do not spin. Do not produce a lengthy explanation of what happened with Thomas Cook and why but if it can be told in short and simple then do, like "Thomas Cook wanted to reschedule but that collided with my daughter's wedding so we agreed on a refund and I found this great deal so I still can have my French-Italian vacation". Frankly: noone cares and it's not relevant. Only state why you changed your itinerary if asked and then state it plain, short and simple. This is key to immigration: never lie to immigration. Never offer information unasked. Stay calm. End.