No problem just change it, you say. But is that all?
The answer to all of these problems are redirects.
Let's say you have this page http://www.example.com/example, and you want to change that to http://www.example.com/other_example.
That will give you all the above problems. But what if you could say: If someone hits the first page (example) then automatically redirect to the second one (other_example). Wouldn't that be great?
Well you can.

You have the site like above example. So now you rename the url fom example to other_example as demonstrated below.

Once you have done that, the old name does not exist anymore. So we are going to create a redirect.
Go to Administer -> Site building -> URL Redirects and click Add redirect on the top of that page.
Hit save and you are done.