While Ishgard has been painted as being on the cusp of change, very little has been stated as far as SPECIFICS of those changes. The only thing I can think of that has been flat-out stated is that Ishgard is now a member of the Eorzean Alliance, and no longer keeping itself neutral in the conflict with Garlemand. To my recollection, nothing has been stated as to whether any House, including Fortemps, has changed their stance on the employment of adventurers and sell-swords.
As for whether the Warrior of Light would be allowed to live within Ishgard/Coerthas, again nothing specific is stated, though it is noteworthy that the guard to the Steps of Faith STILL has the same text blurb indicating that they are allowing you access due to the fact that you're a ward of House Fortemps. Does this imply that you would NOT be granted access, if you were no longer a ward? It's hard to say - but the WoL is pretty much a celebrity in Ishgard, now, and close buddies with the acting leader. I'd guess that if the WoL were to make a request of Ishgard, they would take it very seriously, and if in their power, grant it. Having a home there seems like a pretty small favor to ask. (And if the gate guard WERE to turn the WoL away for some reason, it would likely be a hilarious political faux pass.)
As for social mobility due to dragonslaying, a lot of this is covered in the sidequests at the Convictory in Coerthas Western Highlands - but, unfortunately, it's been too long and I do not remember any details, or whether they'd answer your questions in that regard. Note that the benefits of dragonslaying may be scaled back (ha, "scaled") going forward due to the new peace Ishgard has forged with the majority of dragons. There's still the remnants of Nidhogg's brood to cause trouble, but that's about it. If the peasant or adventurer picked up their dragonslaying cred before the events of Heavensward, though, they could apply.