Is it possible to laser cool without using spatially-varying magnetic fields and above zero degrees? As the title says, is it possible to laser cool without generating a weak quadrupolar magnetic field? Also, if I wasn't trying to achieve extremely low temperatures is it possible to use laser cooling when gas temps are above 0 degrees? If not, why not?
 A: 
is it possible to laser cool without generating a weak quadrupolar magnetic field?

Yes. Optical molasses cooling does not require any magnetic fields. But there is nothing providing a restoring force and hence "trapping"$^\dagger$. Experimentally, you usually first do a MOT and then an optical molasses stage.

is it possible to use laser cooling when gas temps are above 0 degrees

"above 0 degrees" as opposed to what? Negative temperatures?
For laser cooling to have an effect you would need your gas do be isolated from the environment, e.g. in a vacuum, so that it does not rethermalise to the old temperature. Without some sort of spatial-trapping (e.g. MOT), I would assume you could in principle still do laser cooling but you'd need massive laser beams and a lot of optical power.
$^\dagger$: In a magneto-optical trap (MOT), the magnetic fields are not strong enough to trap the atoms magnetically, rather they just modulate the light so the trapping is still provided by a (now modulated) optical force.
A: 
If, for whatever reason, you don't want to use a magnetic field, you can always chirp your laser frequency.

Most ultra-cold atoms experiments starts with a Zeeman slower for a good reason : it's cheap to make and it allows to slow all of the atoms of a given beam with a thermal velocity distribution down to something where usually all of the atoms have a velocity close to $\simeq 10\,\text{m.s}^{-1}$.
The resonance condition between an atom with an eigen-pulsation $\omega_0$ moving at a velocity $v$ against a laser with a wave-number $k$ and a pulsation $\omega$ is :
$$
\omega=\omega_0-k\,v(z(t))+\frac{\mu_B}{\hbar}B(z)
$$
In most situations, the spatial configuration of the magnetic field $B(z)$ is tailord so that such resonance condition is satisfied along the whole trajectory $z(t)$ of any atom. This means that, along this scheme, the cooling process is continuous : at any point $z$ any time $t$, atoms will be cooled down.
If you don't want to use a magnetic compensation $B(z)$, nothing is stopping you from changing/chirping the laser frequency $\omega(t)$ itself along the trajectory $z(t)$ of an atom, such that the resonance condition reads :
$$
\omega(t)=\omega_0-k\,v(z(t))
$$
If this condition is satisfied for any $t$, only a fraction of an atomic beam can be slowed down : only those with a given initial velocity $v_0=v(z(0))$.
Chirped laser cooling as been successfully implemented at various occasions 1 but is rarely used in usual ultracold set-up since it allows to slow atoms only batch per batch (and not continuously unlike in a Zeeman slower), thus leading generally to quite low atomic densities.
1 : Laser Manipulation of Atomic Beam Velocities : Demonstration of Stopped Atoms and Velocity Reversal W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu ;
Phys. Rev. Lett. 54, 996
