I've been surfing the documentation for Splunk DB Connect, and can't find any indication that I need to install anything on my SQL server to utilize Splunk DB Connect.
Is it the case that all installations for drivers, etc., need to take place on the Splunk infrastructure? It seems to me that all you need on the client (e.g., heavy forwarder) are the connection information, valid credentials, and the necessary database drivers.
If a heavy forwarder has those things, it can connect to the SQL server directly, without any additional changes or installs on that SQL server - correct?
it depends on SQL Server version which JDBC driver you must use. In our cases it works also on MS own drivers. See compatibility from here: https://docs.splunk.com/Documentation/DBX/3.5.1/DeployDBX/Installdatabasedrivers#Supported_databases
You must install Splunk DB Connect, along with the associated JDK and drivers, on a Splunk instance - preferably a heavy forwarder. Don't install DB Connect on your SQL server unless you already have a Splunk instance there.
As @richgalloway said you should install it on HF and actually keep it active only in one HF at time. If there is need to migrate it to second HF you must copy also those status files which keep count what events it has gotten already. That for the input/ingesting side.
Over that I suggest you also install it to SH/SHC layer to monitoring those inputs easier as DBX has quite nice dashboards for that. Also if you need to do dbqueries or use dblookups you must install it to SH-layer. BUT don't use those nodes/installations as getting data in to splunk, HFs are for that in distributed environment.
In personally I don't install it to SQL server node even I have HF there already. It's better to keep it on dedicated HF which has used for inputs. And especially if you SQL Server is HA/Cluster, you definitely must install it to another host.