Size Options - Product Variants

The problem is that you created several characteristics with only one value for each.
Instead, you should have created one characteristic with several values.
That should make it way more easy to use for your customers.

For the display, you can change it to radio buttons by changing the option "Characteristics selection method" in the Display tab of the configuration.

Their is a way to have characteristics display or not based on the quantity in stock:
you need to have two characteristic, one characteristic with all your values and one characteristic with only 1 value.
Then, add both to a product without characteristics and change the option "Characteristics selection method" to table.
That should then display similarly to that website.