Now that it is installed, do I need to check for updates my periodically? I can search for "check for updates" and check for updates manually. But will updates (like driver updates, software, or security updates) be checked for and downloaded automatically? I would like it if this were so out of convenience
Sorry but you cannot have it both ways, by uninstalling HP Support Assistant and HP Framework you have uninstalled what is needed for HP to auto update its drivers and software, such software is often referred to as bloatware and with new computers I also tend to uninstall it, there are certain provisos for this though such as what HP hardware a person is using.
If you are only using a HP notebook and everything is working as it should I would be happy as things are, see my canned text below regarding driver updates etc;
Avoid using any software that automatically updates your system.
Please note that drivers should never be updated unless it is to address a known issue, if a driver is installed needlessly you can end up creating the very issue that the updated driver was intended to correct, rolling back to the previous driver does not always work and waiting for the next driver update can take some time and may also not resolve the issue.
Where possible, always obtain drivers directly from the provider of your hardware/software and only use drivers or patches from third parties for older or no longer supported hardware/software.