How to install Craft on live server parallel to existing CMS?
Is the existing CMS live?
Often depends on your whole development process.
Quick thoughts would be to use a subdomain of live site.,e eg 'staging.mysite.com'

For managing redirects the answer is it depends.
Typically htaccess 301 redirects
Redirect 301 /my old url  https://www.mynewurl.com/my old url

You can replace the latter domain name with subdomain name eg as above 'staging' and then switch to real domain when live.

Depending upon Craft version you could use Retour for Craft2 or check the plugin directory for redirect management solutions.

Or if the original url structure is solid then you can always make craft match in section  settings :)