So basically you are trying to make a solar-powered balloon, stretched as a carpet.
Balloon floats because air captured in it is less dense than air around it.
So answering question 3:
- in world with atmosphere with big thermal expansion coefficient. The bigger difference in densities of gases inside a balloon and outside of it, the bigger lift force.
- in world with great gravity. The greater gravity, the greater buoyancy (balloon uses it). BTW. there is no buoyancy in zero-gravity.
- in world with more dense atmosphere higher buoyancy can be achieved (water produces bigger maximum possible buoyancy than air), although air density on it's own does not produce buoyancy (only with correlation with thermal expansion coefficient that produces densities difference).
As to question 2 (I'm not using imperial units here. This is why: https://www.goodreads.com/quotes/8417995-in-metric-one-milliliter-of-water-occupies-one-cubic-centimeter):
Problem with carpet is that most of the heated air beneath it escapes it immediately. I see two possible solutions - capture part of heated up air by shaping your carpet more like a container, or make atmosphere so dense (it will prevent heated air from escaping a bit on it own) with so high thermal expansion coefficient and so great gravity, that this thin layer of heated air would be able to lift whole carpet.
Let's see it it is anyhow doable with some maths.
Some symbols with meanings:
m - mass of carpet
g - acceleration of gravity
Fb - buoyancy force
Vu - volume of heated air under carpet needed to create this force
Rd - difference in densities between heated and cold air (beneath and over carpet) - it gets bigger with air's thermal expansion coefficient growth
Q - gravity force acting on carpet
If your carpet will weight W, then we need to create force greater than W*g to lift it.
This force will be Fb = gVuRd, so:
Fb > Q
gVuRd > m*g
Vu*Rg > m
(Vu*Rd)/m > 1
About thermal expansion coefficient you can read here: https://en.wikipedia.org/wiki/Thermal_expansion#Volume_expansion
This coefficient for gases usually is around 0,01 - 0,02 / K that means, when you heat some volume V of gas by 1 kelvin, you get 1,02V in the end. Let's call this coefficient "B".
This means, that Rd (mo - mass of air over carpet, mu - mass of air under carpet, Rn - density of air over carpet, like in normal conditions):
Rd = (mo/Vo - mu/Vu)
Rd = (mo/Vo - mu/((1+B)*Vo))
Rd = Rn - Rn/(1+B)
So the bigger B, the bigger Rd, and Rd tends to Rn, so to value of density of air above carpet.
So all in all you have:
(Vu * (Rn - Rn/(1+B)))/m > 1
Let's count volume of air under carpet for some example values:
Vu - searched
Rn = 6 g/L (sulfur hexafluoride, one of most dense gases I know about)
B = 0,02 (upper regions of common coefficient)
m - 1000kg (rather small unit assuming aluminum as constructing material, 6 passengers probably)
Vu > m / (Rn - Rn/(1+B))
Vu > 1000[kg] / (0,006[kg/L] - (0,006[kg/L]/1,02)
Vu > 8500000 L (dm3)
So assuming that under your carpet you'll add a wall to keep averagely 10dm (1m) of hot air under carpet (it will be more in the middle, and less on the sides, but I can't calculate this), your carpet would have to have 850000dm2 or 8500m2 of area. Pretty much impossible with 1000kg. But if you'd add a wall that will keep 100m of air under it, it will have to have 85m2 area, so... pretty big carpet you'd have. But possible I think.
Let's mess a bit with entry values:
Rn = 10
B = 0,05
m = 250kg (ultra-light sun-forged meteorite metals alloy)
We get Vu > 525000dm3
So with wall keeping avg. 1m or air beneath, area of carpet should be 525m2 Well.. maybe...?
If we would do:
Rn = 600 (least dense liquid I found has ~616: https://en.wikipedia.org/wiki/Isopentane)
B = 0,1 (this seems so extreme extreme to me that I don't know If it is possible in conditions any carbon life we know of can survive)
m = 10kg
Vu > 1100dm3, so with 0,5dm of air beneath it in average it's area could be 22m2 So extreme...
As to question 1:
Another thing is heating the air - there are two ways that come to my mind. solar powered peltier modules or Stirling engine. Both will transport heat from upper surface of carpet to bottom one. Both can be powered with super-efficient solar power driven energy source, but since this civilization is bound to their sun so tightly, than this is probably not a problem.
Our planes have MUCH more efficient ratio of mass/(wings)area, so maybe just make them all use solar powered planes...?