Good news. I eliminated all the noise!
You won't believe the solution: I loosened the QR skewer of the rear wheel by 1/8th of a turn!
How I diagnosed the problem?
Yesterday, I also tried pushing down on the top tube with me off the bike and the action produced the same squeaks and creaks making me hugely worried the frame is actually cracked. Thinking it might still be the head set, I lifted the front wheel off the ground by pulling on the front of the top tube and then pushing down on the top tube close to the seat tube. I got the same squeaks and creaks which means it's not the headset nor the front wheel.
Thinking of the unlikely possibility it could be the rear wheel, I lifted the rear rear wheel off the ground by pulling at the tip of the rear rack and then pushing down on the top tube near the seat tube. No more squeaks and creaks! First I loosened the QR skewer of the rear wheel and pushed down on the top tube, again no squeaks and creaks. So I tightened the QR skewer again while I'm sitting on the top tube to fully seat rear wheel hub on the dropouts. I used the same QR tightness as before. Squeaks and creaks returned! So I loosened the QR skewer 1/8th of the turn and all the noise is gone!
I've dealt with loose and squeaking / creaking BB cups on the same bike before and the present noise issue sounded exactly like it and the noise seemed like it's coming from the BB shell so unsurprisingly, I thought it's the BB.
Still no time and effort wasted taking the crankset and BB off. The noise seemed like blessing in disguise even though it drove me mad, worrying it could be a crack in the frame. The NDS BB cup I found to be under torqued and 3 of 5 bolts on the chain ring is about to fall off!! I would not have known these problems if I hadn't taken the cranks and BB off because the loose items are not making any noise at all!! Thanks everyone for your participation and patience!