Yes, Ur-Land can be played without having to pay for it. However it is possible, to buy the in-game currency Plombium, with which you can buy supporting items in the In-Game Shop.
You can play Ur-Land also without registration. Registration is required to be able to log in from a second device or to recover from data loss in case of re-installation.
Yes, you can play in as many worlds as you like. Your inventory however is bound to one world, in common is just your username and the amount of Plombium on your account.
To log in to the website, first register in-game. Then, you can choose a username and a password, which will also work on the webpage.
Your email address is just used internally and will not be given to third parties nor be visible somewhere.