Converting heat to mechanical energy Could an object convert heat in the air around it to mechanical energy in order to accelerate itself?
In other words, the craft would take in surrounding air composed of random-velocity molecules and then eject the gas back out in a more unified direction so that it accelerates in the opposite direction.
This would use only that as a power source. So, for example, jet engines wouldn’t work because they use internally stored fuel in addition to outside air. Also I’m really just interested with fluid dynamics in mind. So like no collecting air molecules then fusing nuclei together or anything like that.
Would such a craft be possible? Or would some principle get in the way. For example, maybe the drag caused by collecting the gas would always cancel out the force caused by ejecting it, or something along those lines, I don’t know.
Please let me know if I should phrase the question more clearly.
 A: 
Could an object convert heat in the air around it to mechanical energy in order to accelerate itself?

Only if the engine contains a heat reservoir at a lower temperature than the surrounding air (and even then, only for a limited period of time). For example, it could contain a tank of cold liquid with a low boiling point, use the atmospheric heat to boil this liquid, and use the expanding vapour to move pistons, drive wheels etc. (basically a low temperature steam engine).
Without a low temperature heat reservoir, this would break the second law of thermodynamics. Specifically, it would contradict the Kelvin-Planck statement of the second law which says

It is impossible to devise a cyclically operating heat engine, the effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work

A: 
Could an object convert heat in the air around it to mechanical energy
in order to accelerate itself?

Yes, the object (heat engine) can theoretically convert heat entirely into mechanical energy in a process. An example is the reversible isothermal expansion process in the Carnot Cycle.
But the heat engine cannot convert heat entirely into mechanical energy while operating in a cycle. That would violate the following Kelvin-Planck statement of the second law (bold face emphasis mine):
No heat engine can operate in a cycle while transferring heat with a single reservoir
To satisfy the second law some part of the heat taken in from the air would have to be rejected to a lower temperature environment. The maximum possible cycle efficiency is that for the Carnot Cycle.

In other words, the craft would take in surrounding air composed of
random-velocity molecules and then eject the gas back out in a more
unified direction so that it accelerates in the opposite direction.

What you are describing here is not converting heat into mechanical energy as you initially stated in the first sentence. Heat would be energy transfer from the air to the object due to temperature difference. What you are describing is the transfer of mass (air) to the object and the object somehow taking the internal kinetic energy of the molecules of air and using it to propel the object. The problem is you have not described a mechanism for doing it and should do so.
Hope this helps.
A: What you have described is a variation on Maxwell's demon which was proposed by the physicist James Clerk Maxwell in 1867. Quoting from Wikipedia:

In the thought experiment, a demon controls a small massless door between two chambers of gas. As individual gas molecules (or atoms)
approach the door, the demon quickly opens and closes the door to
allow only fast-moving molecules to pass through in one direction, and only slow-moving molecules to pass through in the other. Because the kinetic temperature of a gas depends on the velocities of its
constituent molecules, the demon's actions cause one chamber to warm
up and the other to cool down. This would decrease the total entropy
of the two gases, without applying any work, thereby violating the
second law of thermodynamics.

In your case, if it were possible, the temperature difference between the two chambers could drive a heat engine and, say, a propeller. The quoted wikipedia article also mentions a 2006 paper describing using a laser to sort molecules however:

the process requires more energy from the laser beams than could be
produced by the temperature difference generated.

so the laws of thermodynamics are not violated.
