For the lining usually 3-4mm is used – the thinner stuff is lighter and fine if you take the care to secure things in the back – the thicker stuff is good if you will treat the interior like a tradesman. You can get pre-cut kits – google is your friend.
9mm minimum for the floor – 6mm will crack up.
Insulate with sheepswool – smells it a bit for the first few day but it will get better with time and its so much nicer to work with than glass-fibre and you won't be inhaling fumes as foams degrade. Check this video out for helpful tips. you can buy the insulation kits from amdro too. There are other useful vids on his youtube channel.
Covering the ply floor in contract vinyl flooring is a good idea.
LED's are good, wire them to the LB that is charged via a split charging relay taken off the engine battery with the signal wire taken off the alternator. Remember to fuse it all properly.
Leisure batteries come in all shapes and sizes – find one that fit the space you have.