How can I downgrade my MacBook Air from macOS Catalina to Mojave?

I have nothing on this MacBook, no files whatsoever. Do I still have to do the backup?

Since there is nothing on that MacBook that you need, no. The recommendation to create a backup first is that (believe it or not) some people do not understand erasing the Mac means that all its files will be gone.

Will I be able to turn it on again without doing backup?

Assuming the Mac is otherwise operational, yes, you should be able to. However, the original stated problem is that the Mac is performing poorly. macOS upgrades don't cause that, meaning your proposed "downgrade" is not likely to fix anything anyway. Before considering doing something that drastic, download EtreCheck and post its report in a reply. Perhaps it will suggest reasons that Mac isn't running well.

Instructions: How to use the Add Text Feature When Posting Large Amounts of Text, i.e. an Etrecheck Report - Apple Community

Nadziejkax wrote:

I’ve MacBook Air mid 2012 I think and I upgraded the system to Catalina but it’s SOOO slow I cannot function like this. [...] Will I be able to turn it on again without doing backup?

Does that Mac have internal HDD or SSD? Booting from an APFS HDD is very slow and Mojave and later need APFS. High Sierra is the latest to support booting from MacOS Extended which is OK on a HDD.

I'd clone the internal boot volume to a bootable external disk with Carbon Copy Cloner AND also create a bootable macOS installer before erasing the internal disk. That makes life easier if poking the internal disk makes it and its Recovery volume unbootable.

Create a bootable installer for macOS - Apple Support