Can 10.12 and 10.13 coexist on one Mac?

Can 10.12 and 10.13 coexist on one Mac?

Yes. Partition the disk using Disk Utility, and install macOS on the newly created partition.

But knowing the 10.13 has a new filesystem will my Mac work in 10.12.6 with no problems when I install 10.13 on another (experimental) partition?

Yes.

Will the installing change all my other disks to the new filesystem?

No.

Does the installation change anything in the computer itself (I mean any firmware change disabling earlier systems?)

Any required firmware updates will not disable other operating systems.

Just a couple of notes:

Unless your hard drive is an SSD, 10.13 will not automatically format it as APFS. If it is an SSD it will only format the partition or volume its being installed on and not touch any of the others.

While 10.12 Sierra will still be able to see and access the APFS partition and file system for 10.13 if formatted APFS, 10.8 and 10.10 will not. It will simply not be visible or usable by them normally if formatted as APFS.

You cannot boot into an operating system that is older than the one that originally shipped with the Mac when new.

Another alternative is using Parallel's Desktop Lite (free, Mac App Store) to contain multiple OS X/macOS guests on an external, USB3 SSD drive. Ironically, the El Capitan 10.11.6 guest boots faster, than the iMac (late-2013, 250GB SSD) that is running High SIerra 10.13.6. I haven't updated my RAM to 16GB yet, but is is imminent. The Parallel's application will use the Apple .dmg installer that it detects.

Had you mentioned Windows originally, or gaming in your multisystem reference, then I would not have posted, because the free Parallel's Desktop application cannot run Windows as a guest without a $60/yr subscription, and gaming in a virtual machine is contraindicated.

My response was simply an alternative augmentation to John Galt's response.