Wheel squeaking at slow speeds and slight play in the wheel bearings, related? When I am pushing the bike or (lately) cycling slowly, I can hear/feel a slow squeak/skreech from the front wheel. I can feel it through the handle bars, and it isn’t quite the sound of brake pads rubbing. A while back when I started noticing this noise, I only really heard it when pushing it backwards or when pushing it at an angle. I also noticed that the front wheel has a bit of left/right play when I hold it and push it side to side. It does not make any noise, and rotates freely, when lifted in the air and rotated. 

Are the play in the wheel axle and the noise related? Do I need to tighten the cones? Or should I grease them first?
If there is noticeable play in the wheel bearings, they are worn or need adjusting. Noise is most likely related to this.

The difference when rotating the wheel forward or back, or off the ground is due to the different loads on it.

You first need to figure out what kind of bearings you have in your hub. Cartridge bearing should just be replaced. Cup and cone style can be disassembled re-greased and adjusted (or replaced if they are worn out).