Raspi is definitely a good starting place for Home Assistant (the Hassio distro makes Home assistant particularly easy to pick up). If / when you grow out of it, an older laptop or desktop would probably be more than enough for you. I’m a big fan of Home Assistant!

Are the integrations with non-branded off the shelf hardware (e.g. 433Mhz Gas sensor) instead of branded well known IoT sensors possible?

I integrated all my 433MHz devices by listening for them on a Sonoff RF Bridge (https://sonoff.tech/product/accessories/433-rf-bridge) which then pushes the signal to MQTT. From there it is easy to pick it up.

Generally speaking the main problem is the RF part (to be able to catch it somewhere, and reuse)

I should have mentioned that I'm using RTL-SDR to catch the signal.

And how do you interface to it? Do you send what you caught to MQTT?

I am considering having something else as the RF receiver but would need to forward what it received to MQTT (I can write that part myself - it is really the interface to the dongle I would like to understand)

I use DVB-T USB dongle. You can use this project[1] to decode it.

[1]https://github.com/merbanan/rtl_433

It is. Often via simple configuration.