shared code between many lambdas

I would not use a separate Lambda function just to create the connection. I am not even sure if it is possible. The other function runs in its own isolated execution environment so you can't share the connection.

You should create the connection in each Lambda function that needs it. You can use a Layer, but I prefer to just include the code in the function and call it where you need it.

The first option is an absolute no-no.

Take a look at this tutorial for an example - https://docs.aws.amazon.com/lambda/latest/dg/services-elasticache-tutorial.html

Make sure that the connection opening code is outside the event handler code in the lambda, otherwise a new connection will be opened with every event.

If you are concerned about high concurrency of lambda causing a high number of connections to Redis, you can set reserved concurrency on the lambda so the number of connections does not go beyond what your Redis cluster can accept. The clients will then have to implement retry in their code to handle throttles due to reserved concurrency on the lambdas.