If you don't want to change the height of the floor with a traditional rubber membrane, you can use a waterproofing/isolation membrane called RedGard. It is applied like paint, and only would add a fraction of an inch to the height of your floor.
The manufacturer recommends that you put on one coat to act as a crack isolation membrane, and 2 coats to be waterproof. Since this is an on grade slab, it wouldn't hurt to add a second coat to prevent water from being absorbed from the ground under your tiles.
A crack that size will surely telegraph into the tile. Before you apply the membrane, you should fill the crack with hydraulic cement or mortar. After the crack is filled, you can use your level to find any high spots, and then grind them flat with a grinder that is equipped with a diamond cup wheel.
Since this is in the basement, it would also be a good idea to use a mold resistant grout. This will keep the floor looking new for a long time.
Those tiles are going to be ice cold in the winter, and I also don't see any other heat source in your picture. One other thing you could look into is an under tile heating system. They can be a little on the expensive side, but would be perfect for your bathroom. Some of them also act as an isolation membrane too, so there will be less of a chance of your tiles cracking.
One last thing to consider is installing a stone threshold. Even if you don't change the height of the floor, you can get one that is the same thickness as your wood floor and the tile. It will give you a much cleaner transition from the tile to the wood. There are also a wide variety of thresholds available on the market which are designed to overcome height differences. If you are concerned about tripping over it, there are ones that are beveled. You shouldn't have a problem finding one that works for your bathroom.