Would hot hydrogen (in the same sense as hot air) be able to lift even more mass?
Yes. Though I suppose the fire danger goes up, and you certainly can't use a propane burner to warm it...
Would a higher or lower density of hydrogen in a ballon lift more?
Lower density always means higher buoyancy.
If you could have a balloon which had nothing in it (it was a vacuum inside) would that lift more than a hydrogen balloon?
Yes, and this has been proposed in various ways in science fiction literature. The engineering challenge is finding a away to confine the vacuum that is as light as a gas bag so that you don't loose the advantage to extra weight.
In general a volume $V$ of material of density $\rho$ immersed in a fluid of density $\rho_f$ experiences a buoyant force of
$$ F_b = gV\rho_f $$
and a weight of
$$W = -gV\rho $$
so the available lifting force is
$$ F_l = gV(\rho_f - \rho) .$$
Where the object is floating at the surface of a liquid the buoyant force is modified to reflect the volume of liquid displaced $F_b = g V_d \rho_f$ where $V_d$ is enough to cover the weight of the floating object.
Balloons are buoyant because the air pushes on them. The air doesn't know what's in the balloon, though. It pushes on everything the same, so the buoyant force is the same on all balloons of the same size.
If the "balloon" is just a lump of air with an imaginary boundary, then the lump won't go anywhere because the air isn't moving on average. So the buoyant force must exactly cancel the gravitational force (the weight). Since the buoyant force is the same on everything, the buoyant force on a balloon is equal to the weight of the air it displaces. In symbols this is
$$F_{buoyant} = \rho g V$$
where $\rho$ is the density of air, $g$ is gravitational acceleration, and $V$ is the balloon's volume.
Hydrogen and helium have less weight than a similar volume of air at the same pressure. That means the buoyant force on them, which is just enough to hold up air, is more than enough to hold up the balloons, and they have to be tethered down.
Assuming they have the same pressure and volume, a hydrogen balloon has less weight than a helium balloon. Things like pressure and volume are roughly decided on a per-molecule basis, at least in gases at low pressure, so at the same pressure and volume hydrogen and helium will have the same number of molecules. Hydrogen is lighter per molecule, so the hydrogen weighs less, and less of the buoyant force is canceled out. That means the net force on a hydrogen balloon is greater. The difference in the net force is small.
Hydrogen is $H_2$, which has atomic mass 2, while air is mostly $N_2$, which has atomic mass 28, so the hydrogen balloon has a net force of about (28-2)/28 = .93 the weight of the air it displaces.
Helium is mostly helium-4, so the net force on a helium balloon is about (28-4)/28 = .86 the weight of the air is displaces.
So net force on the hydrogen balloon is in the neighborhood of 10% more.
Best Answer
You need to take density of the air into the question. And weight of the balloon itself.
The Archimedes' law says:
Therefore a balloon can support the same weight as equal volume of air has (reducing gravitational acceleration from the equation as it appears on both sides). Which includes weight of the baloon itself. And note that weight of the rubber will probably be considerably higher than weight of the helium inside.