Will I run into problems if I install an additional site with CMS version 5.2.375.236 on a server currently running CMS version 5.2.375.133?
Nope. Those versions should work great side by side :)
When contacting the hosting company they say that there is no deployment manager to be found on the server, it seems to have been removed for unclear reasons.
How can a new deployment manager be installed on the server without causing any disruption or down time for the current site?
Installing the deployment manager and just installing the Application files should not cause any down time for the current site. Not sure how the installer behaves if the EPiServer files are on the server before running the installer. (try it on your computer)
Overwriting the EPiServer files is the only impact the installation might have on the running site(s). And that only impacts the admin/edit/util files. No guaranties given, but personaly I would go ahead with the installation.