How am I supposed to use a Postgresql docker image/container?

1.

docker run --name some-postgres -e POSTGRES_PASSWORD=mysecretpassword -d postgres

After I stop the container I cannot run it again using the above command, because the container already exists.

Correct. You named it (--name some-postgres) hence before starting a new one, the old one has to be deleted, e.g. docker rm -f some-postgres

So I start it using docker start instead of docker run. Is this the normal way to do things? I will generally use docker run the first time and docker start every other time?

No, it is by no means normal for docker. Docker process containers are supposed normally to be ephemeral, that is easily thrown away and started anew.

2. Persistance: ... I can stop and start the container and the data is persisted, although I'm not sure why or how is this happening. ...

That's because you are reusing the same container. Remove the container and the data is gone.

3. Architecture: from what I read, it seems that the most appropriate architecture for this kind of app would be to run 3 separate containers. One for the database, one for persisting the database data, and one for the node app. Is this a good way to do it? How does using a data container improve things? AFAIK my current setup is working ok without one.

Yes, this is the good way to go by having separate containers for separate concerns. This comes in handy in many cases, say when for example you need to upgrade the postgres base image without losing your data (that's in particular where the data container starts to play its role).

4. Is there anything else I should pay atention to?

When acquainted with the docker basics, you may take a look at Docker compose or similar tools that will help you to run multicontainer applications easier.

Short and simple:

1. What you get from the official postgres image is a ready-to-go postgres installation along with some gimmicks which can be configured through environment variables. With docker run you create a container. The container lifecycle commands are docker start/stop/restart/rm Yes, this is the Docker way of things.
2. Everything inside a volume is persisted. Every container can have an arbitrary number of volumes. Volumes are directories either defined inside the Dockerfile, the parent Dockerfile or via the command docker run ... -v /yourdirectoryA -v /yourdirectoryB .... Everything outside volumes is lost with docker rm. Everything including volumes is lost with docker rm -v
3. It's easier to show than to explain. See this readme with Docker commands on Github, read how I use the official PostgreSQL image for Jira and also add NGINX to the mix: Jira with Docker PostgreSQL. Also a data container is a cheap trick to being able to remove, rebuild and renew the container without having to move the persisted data.
4. Congratulations, you have managed to grasp the basics! Keep it on! Try docker-compose to better manage those nasty docker run ...-commands and being able to manage multi-containers and data-containers.

Note: You need a blocking thread in order to keep a container running! Either this command must be explicitly set inside the Dockerfile, see CMD, or given at the end of the docker run -d ... /usr/bin/myexamplecommand command. If your command is NON blocking, e.g. /bin/bash, then the container will always stop immediately after executing the command.