So I'd like to ask, if your Warden is alive, what do you headcanon them doing during DAI and after DAI?
Watching. Simply put, she's not religious in the slightest and is very Anti-maker. During the events of the Inquisition she's started to come across as quite 'evil', she very much sees the Inquisition has serving a purpose but she doesn't like them - she'll end them if given the chance. I do see her deeming it necessary to intervene in the Orlesian warden crisis, if only to save the Ferelden Wardens from the same fate, if only to prove a point. (Dat power grab)
Unfortunately "helping" for her isn't so easy because she didn't exactly follow the God-Baby canon, she did something more akin to Flemeth. >_> (Get rekt Morrigan)
Suffice to say my warden wasn't a big fan of the scheming witch of the wild's or the "psychotic, religious freak".
Disclaimer: I do love Leiliana, she develops really well in Inquisition even if I was irked that my decision to kill her in my playthrough was retconned lol.
Do you follow Bioware's guidelines and have them search for a cure or do you have them doing something completely else?
Nope! My HoF, as I said, is "sleeping". She's a mage folks, people would be foolish to assume that Morrigan/Flemeth/Solas are the only ones who delved into deeper magic.
If they're doing anything in particular, do they have any specific reasons or motivations to do this?
Sleeping and regathering her strength, although for the most part she doesn't have a choice in this (technically).
Did/do they have direct contact with the Inquisitor or Hawke? If yes, what's their relationship like?
Yes and No.
Hawke: She merely watched, I have two wardens (An Orlsesian for Awakenings, and a Ferelden for Origins). While the Orlesian warden should have slapped Anders senseless, the Ferelden warden is still very much an observer. Pro-tip, she found my Hawke extremely weak and much too benevolent, she firmly believes if Hawke had more backbone he would have stopped Anders from being so utterly stupid, to her there were better ways of destroying the Chantry. Granted, she also places a great deal of blame on my Orlesian warden for failing to control and oversee his own 'troops'.
Orlesian Warden: Weak, weak, weak, weak, weak! HoF does not like Orlesian warden, much to weak, disorganised and naive to politics! Orlesian warden was just an ordinary soldier thrown into chaos, it wasn't his fault.
Inquisitor: She starts by observing until she's "released" from where she's sleeping... Then she spends most of her time taunting an order she very much considers to be fruitless (again, remember she's not religious). She's ultimately waiting for the Inquisition to "fail", though she seems inclined to let them throw body upon body at Corypheus until he dies/is weakened.... Overall, my HoF aims to destroy them later, but for now they serve a purpose (she's so nice right

)
The relationship the HoF and Inquisitor have is a precarious one, Leiliana has warned of the HoF's demeanour and my Inquisitor agrees that she's highly suspicious... But she's a mage, and a powerful one, so better to 'watch' and 'use' her to their advantage.
Oddly she seems unconcerned with the blight that should be running through her veins, I can't ever think of why.