(Heroku + Sidekiq) Is my understanding of how Connection Pooling works correct?

I don't know Heroku + Rails but believe I can answer some of the more generic questions.

From the client's perspective, the setup/teardown of any connection is very expensive. The concept of connection pooling is to have a set of connections which are kept alive and can be used for some period of time. The JDK HttpUrlConnection does the same (assuming HTTP 1.1) so that - assuming you're going to the same server - the HTTP connection stays open, waiting for the next expected request. Same thing applies here - instead of closing a JDBC connection each time, the connection is maintained - assuming same server and authentication credentials - so the next request skips the unnecessary work and can immediately move forward in sending work to the database server.

There are many ways to maintain a client-side pool of connections, it may be part of the JDBC driver itself, you might need to implement pooling using something like Apache Commons Pooling, but whatever you do it's going to increase your behavior and reduce errors that might be caused by network hiccups that could prevent your client from connecting to the server.

Server-side, most database providers are configured with a pool of n possible connections that the database server may accept. Usually each additional connection has a footprint - usually quite small - so based on the memory available you can figure out the maximum number of available connections.

In most cases, you're going to want to have larger-than-expected connections available. For example, in postgres, the configured connection pool size is for all connections to any database on that server. If you have development, test, and production all pointed at the same database server (obviously different databases), then connections used by test might prevent a production request from being fulfilled. Best not to be stingy.