Are they waterproof? Like enough to be worn outside?

They are somewhat good at keeping with elements but really wet weather not so much but worth the price..

What are the materials used?

The material is very durable and stretchy creating a comfortable and washable long lasting shoe that can always just turn into play shoes if nothing more. I absolutely love the material and durability of this shoe and have bought no other kind but these since she was 8 months old.

What is the size of the 0-6 month? How many cm or inches?

In my opinion I’ve had 8 babies lol at that age kids hate shoes an it really can be a waste of money. However depending on your babies foot I would say sizes go according an if it’s 6 months they say it. That is another reason I love these shoes my daughter has a chubby foot and all my 7 sons hated shoes but she loved these. They fit more like socks and are ready for on the go type lifestyles. I know this is sort of vague but trust me these type shoes are a god sent.