Oops - just answered the first part of your question there sorry.
The way I tanned the chicken skin is a form of vegetable tanning. Large tannin molecules from plant material (ie bark from trees such as mimosa, oak, chesnut or willow) bind with the proteins of the skin and because the tannins are very stable, they prevent rotting.
Brain tanning is a bit different. Basically it is a form of "fat liquoring", where fats from the brain surround the protein fibres and preserve them. With brain tanning however, if the finished leather gets wet, it will go stiff. To help prevent this the skin can be smoked. Chemicals in the smoke react with the brain fats and add a further stabilising effect. Brain tanning produces beautiful soft buckskin leather (after a lot of work). Vegetable tanning produces a harder leather, such as might be used for belts, knife sheathes etc. Most modern leather is chrome tanned. This can get wet and be dried again with no ill effects, but the leather cannot be moulded or tooled and blunts your tools very quickly. Chrome salts are also toxic, so its not a method to try at home.
Thin skins like chicken and rabbit can also be preserved by rubbing alum into them. Alum can be got hold of quite easily as it is used as a mordant in dying cloth.
In short, there's a lot of options depending on what you want the leather for. Possibly your easiest bet if you want to try a chicken skin is to use willow or alder bark. Strip off about a carrier bag full of bark in March/April time, break into small pieces and boil up in water for about two hours at a slow simmer. Put a dab of the cooled liquid on your tongue. If it feels like drinking really strong tea it should be OK. Strain off the liquid and put it in a plastic bucket with your chicken skin. Stir daily. After a week, cut a small slit an edge of the skin to see if the skin is evenly tanned. If there is still a pale line in the centre, put back in the tanning solution (timing depends on the strength of the liquid, thickness of the skin and the temperature ~ slower if very cold). If it is done, take out the skin and wash it well, put in water with a little vinegar in over night to "fix" the tannins. Allow to dry until just still damp and rub in oil ( fish oil is traditional, but neatsfoot will do or even olive oil at a pinch). The skin should now be ready to use.
Hope this hasn't bored you rigid and good luck.
Willowbark.