At first we should distinguish which fighters are meant for which carrier and also why the fighters IN ordered so far, were procured!
INS Vikramaditya was bought as a combined deal, the carrier basically for free, but we had to buy 16 x Mig 29Ks + pay for the refurbishment of the carrier. IN in that time even evaluated the Rafale M and asked Boeing if the SH could be operated from that carrier, but since we had to buy the Russian fighters and the US didn't wanted to provide us catapults for IAC1, there was no other choice than the Migs.
Admiral (retd.) Arun Prakash was the first Indian that has flown in a Rafale:
IAC 1 as mentioned had to be designed as a STOBAR carrier similar to INS Vikramaditya, which made it logical, that we use the same fighters on both carriers for ease of operations and commonality. That's why the additional order of 29 x Mig 29K was placed, which will be complemented with the N-LCA later and that leads us to this famous interview:
LCA-Navy Not What We Want, But It's Ours
...The officer's response: "I wish wish we could straightaway develop a Rafale. But seriously, we have to look at the Indian Navy and it commitment towards indigenisation...
Livefist: "LCA-Navy Not What We Want, But It's Ours": FONA
Again a clear sign that IN would love to have Rafale M, but again for different reasons, they had to take another fighter, in this case the limited capable N-LCA, just for supporting the indigenous industry in developing an Indian carrier fighter.
IAC 2 is planned for the end of this decade and for this carrier IN sent out the RFI for 40 new carrier fighters. The RFI was send to Mikoyan, Boeing, Dassault, LM and the EF consortium, so basically any manufacturer that had at least a study of a carrier fighter and (by then) was involved in the IAF MMRCA competition. Saab asked IN/MoD to join the naval competition as well and now started to develop the Sea Gripen with Brithish help.
Why is this RFI important? Because if we develop a new carrier, we have to know what kind of fighters it will use from the start as well. We have to know the size and the weight of the fighters, because they have to suite the hangar, elevator and arrested landing limitations. The same is also important for the layout of the carrier and which fighter take off system it uses, ski-jump or catapults. So when we get the infos from the vendors, we can check which of them suits at best to the specs of IAC 2 that IN is planning with.
Brazil for example, did not only evaluated Rafale, Gripen and F18SH for the airforce, but also if they fit to their current carrier (Sao Paulo, ex French Foch) and those they plan for the future. Their current carrier has catapults but still, that doesn't mean any fighter can be used from it the same way. Reports said, that the Sea Gripen can be used from it with it's size and weight, Rafale M can be used from it as well (the French even tested it in the past), but it could only take off with limited payload. The F18SH instead couldn't been used because of the heavy weight and size, that would have been even too much for the elevators. So even if Brazilian air force would go for the F18, the navy has to buy a different fighter for their carriers and IN will do exactly the same evaluation, with the difference, that they has no limitation of a current carrier, they can develop the future carrier around the fighters they will select and a common choice to IAF of course give some advantages.
IF Rafale wins IAF MMRCA, it would be a very good choice for IN as well, but just like it was the case in the past, I think this time again a combined deal might force the navy to procure an US fighter, be it the F18SH, or the F35C. IN wants catapults, if possible even the new EMALS system, but the US might offer them only with a fighter deal, or we will be left with a ski-jump layout again, which makes IAC 2 to comparable by size and layout to the Russian Admiral Kuznetsov class of carriers.
Regarding carrier fighters, the only differences that makes a carrier fighter suited for catapult take offs are, more strengthened airframe and front gear. Besides that, they are not different to the a carrier fighter used on a STOBAR carrier, that's why Saab offers the same Sea Gripen for STOBAR and CATOBAR carriers.
The issue with Rafale taking off from ski-jump carriers imo is only the thrust, since a good TWR is needed, because the take off is unassisted. The Mig 29K has a better AB thrust than Rafale M, so that could be a reason, but otherwise there is hardly a reason.
Regarding N-Pak Fa, the Russian offered us to lead the re-design to a naval fighter and let HAL do it (obviously to safe some of their money), but a naval version is possible and since future Russian and Indian carriers are aimed to be around the Admiral Kuznetsov class, that already operates Flanker class fighters, why shouldn't a naval Pak Fa not fit on IAC2?
AFAIK IN only rejected the idea of naval Su 30 MKIs (better Su 33 MKIs) on INS Vikramaditya or IAC 1, since these carriers are too small for such fighters.