I have heard not to caulk around the batten so moisture can release Incase it gets trapped inside, but also heard caulk all around battens also to block and moisture going in.

I doubt that too; maybe it's just me but it puts me off to see carpet in a bathroom situation and in a place that may or may not have moisture issues - I'd just as soon see the floor painted and put down an area rug. 

There are some small cracks in the floor and most of the moisture was around those cracks kind of like water is coming up through the floor a little bit.

Those point to the need for mold remediation, moisture and/or water control.Is there knob & tube wiring? 

Tenant shall be held responsible for water damage caused by overflowing toilets and/or drains.Take no action which would cause a cancellation or increase in Landlord’s fire or liability insurance.Obey all environmental laws and keep no toxic, inflammable or dangerous substances on the Premises.Prevent the accumulation or buildup of mold or mildew throughout the Premises by notifying Landlord immediately if any evidence of water leak or excessive moisture, mold or mildew appears.

Too much humidity and moisture for the pressure sensitive glue on the back of the tile; too easy for glue to fail and don't even think about adding a glue to help it stick.

@Bryan Devitt The attic is sort-of finished and home inspector thinks there is no good air circulation and moisture which causes mold to form.

It does not tolerate moisture and swells up at the seams instantly if exposed to it. 

Cracks in the foundation, dangling wires, three layers of roofing material on the roof, broken fixtures, off color carpet areas, spots on the ceilings, soft spots in the walls in the bathroom where water has penetrated, swampy areas in the lawn.

Often adding a downspout extension to direct the water from the gutters away from the house will help and is easy to do.For the moisture, they should do whatever they can to dry it out quickly.