It seems boot loaders from the syslinux family aren't able to boot from another drive as the one the boot loader is installed on (I'm not sure; I'm more used to grub2 where it is possible).
About installing TC on the same partition. I don't have a problem installing it there;
but will TC be in it's own separated folder?
Yes. on sda1, it will be /mnt/sda1/tce/.../
Will I be able to later remove one of them and keep the other, just by deleting it's folders?
Yes. But make sure you don't delete the directory which contains additional boot loader files and the config file (either /boot/extlinux/ or /tce/boot/extlinux/).
And if the same will happen - will I be able to delete TC's folders with Puppy's live-CD; and the problem will go away?
I don't understand. If something doesn't boot, you probably won't be able to fix it by removing stuff. You'd rather make it worse if you're not absolutely sure what can be deleted.
It'll really be a problem if I won't be able to boot into anything, and will have to reinstall and loose all the settings (mostly FireFox).
The good thing about, both, Puppy and Core, is that you can easily backup your settings to an external drive (Flash, CD, whatever) in case something terrible happens. Both distros save the user settings in packages.
For Core it is
/tce/mydata.tgz
For Puppy I'm not sure right now but usually it's something like
/puppysave.sfs