Mongoose is an event-driven networking library including HTTP server and client, MQTT server and client, SNTP client. It runs over many platforms that provide a Berkely sockets compatible API, including lwIP, though nowadays (v7.7+) it also has its own TCP/IP stack to run on baremetal environments.
It has plenty of examples and tutorials, and besides the fact that Google is your friend: https://mongoose.ws/ ← you’ll find links to both, and the docs, there
Mongoose-OS is an event-driven framework that uses Mongoose v6.? as a networking middleware. It runs over several platforms and leverages vendor drivers (e.g.: Espressif SDK), usually providing friendly event-driven APIs on top. Again, Google is your friend but Mongoose OS - an IoT Operating System (the graphic there is probably quite more precise and enlightening than my words here)
You can use all APIs simultaneously, you can call Mongoose-OS API functions, Mongoose API functions, ESP IDF functions, FreeRTOS functions, use the hardware directly, there is no MPU nor firmware police preventing you from doing what you want to do. A particular example of using Mongoose directly is an HTTP client, though you could take a deep breath and use ESP-IDF for that, no one will blame you for that.
You can’t use different versions; each Mongoose-OS version is pinned to a Mongoose version and a vendor SDK version (e.g. ESP-IDF)
IIRC there is a version of Mongoose-OS that runs on Linux, so you could run it on Docker. You won’t have chip (ARM/ESP32) simulation, though
Mongoose-OS uses YAML to simplify building process and pinning to particular libraries as it is a very complex piece of software. Nothing is usually “necessary”, though you’ll have a hard time doing everything manually.
Mongoose is amalgamated to just 2 files and you put them wherever you want and reference them on your IDE of choice (or command line of choice). Examples use make 'cause it is readable by most users, including make-illiterate developers like myself.
You have dozens of Mongoose examples in the Mongoose site and dozens of examples in the Mongoose-examples repo.