First of all, I’m sorry for this possibly extremely dumb question. 
Like this page told, I removed all the 13.1 specific repositories and replaced them with Tumbleweed ones. What then? Am I now on Tumbleweed? Is it just a collection of repositories or do I have to do something else? I’m asking this because I was under impression that Tumbleweed was completely different derivative of openSUSE.
Thank you in advance for any answers, and again, I’m sorry if this was a stupid question.
If you added the Tumbleweed repository and did a “zypper dup”, then you are “on Tumbleweed”.
Tumbleweed is not a derivative, but an add-on repository, delivering newer versions of some software of openSUSE “current” ( = 13.1 atm).
And especially new Linux kernels are delivered often.
Hendrik
After replacing the repos, you have one new Tumbleweed repo, the other repo addresses actually point to the standard openSUSE repos for the current release (13.1). It means you don’t need to edit the repo addresses when Tumbleweed re-bases on the next release (13.2 much later this year.) The Tumbleweed project maintainer switches to the appropriate repos when the time comes.
Am I now on Tumbleweed? Is it just a collection of repositories or do I have to do something else? I’m asking this because I was under impression that Tumbleweed was completely different derivative of openSUSE.
Did you continue with the instructions on that page, i.e. perform a distribution upgrade using the “zypper dup” command in a terminal? If “no” is your answer, then you have not yet installed Tumbleweed.
Yes, after installation/upgrade, it is a different derivative of openSUSE, and the differences in packages lie mainly in the Tumbleweed repo itself.
If you use third-party repos, Packman for example, then you will need to add and enable the Tumbleweed versions of those repos. You can also do that after the first upgrade to Tumbleweed, if you didn’t include them originally. This openSUSE Wiki page explains third-party repo additions: http://en.opensuse.org/Additional_package_repositories
For regular maintenance, you should always update Tumbleweed using zypper dup, that will include all security updates, package upgrades, and fixes (patches). That is the only supported method.