You would need full FTP access to the site along with logins for any databases you may have.
Ideally you would have access to the hosting control panel as all this information would then be available to the new designer.
You should also have full control over your domain name records...
As soon as you receive any log in details you should change the passwords. I know of someone (no me) who fell out with the designer over the final bill and the designer changed all the passwords. Took weeks to correct.You would need full FTP access to the site along with logins for any databases you may have.
Ideally you would have access to the hosting control panel as all this information would then be available to the new designer.
You should also have full control over your domain name records...
Depends on the ecommerce package you use. If it's an off the shelf application then you need to make sure you have the licence. If it's a bespoke design then it's unlikely that the current designer would release the code and a new designer be able to do the developemnt you need.
As to choosing a new designer, you need to detail the ecommerce package you have, in particular if it is PHP or ASP (or something else) as not all developers work with both. Or maybe you don't need the backend changing just the template or the images or the navigation. Each may require different skills set so the brief is quite important.