I wouldn’t recommend doing so. It would be better to change the subdomain from www to cloud or something else.
Not sure, but I’d say you would have to move the nextcloud folder to a subfolder of your current webroot, if you absolutely want to do it like that. But as I already said, I higly recommend to use different subdomains / webroots for different services, instead of multiple subfolders under the same (sub)domain / webroot.
Yes, in addition to changing the configs (Apache and Nextcloud config.php), you would also have to change your existing DNS record from www to whatever subdomain you want to use. Also if you are using signed SSL certificates, from e.g. Let’s Encrypt, you have to re-issue them for the new subdomain, unless you are using a wildcard certificate.