Can my first datacard be a choice field changing which data cards are displayed on a PowerApp form?

Yeah, you would have to have all your datacards just reference the value with an if() statement on the visible property of the card.

So Visible = if(datacard1.value = yes,true,false) etc. If alignment doesn't seem right you might have to check on doing the same inside the positioning of the field etc. But it is possible. If you can elaborate more on it, I would suggest taking it over to the actual powerapps community as there are tons more Poweraddicts there willing to help that know more than myself :).
https://powerusers.microsoft.com/t5/Creating-Apps/bd-p/PA_Creating_Apps