Sounds like a fun idea! The fact that this isn't a one/two day event makes it a lot easier for me to participate 
That's good to hear.
Sounds great! Where do I sign up? (Seriously, do I have to register on that site or something?)
It might be a good idea to! Once you're signed in, there's a spot to sign up officially on the event page, but it's not compulsory. There might be ways around registering if you were really opposed to making an account, but given that Glorious Trainwrecks has been hosting Knytt Stories related events for 6 years now it will probably pay off.
How big should a half-finished level be? Are there any rules?
Any size you want, but smaller is probably better. A week is just a convenient time period, don't feel like you even have to use it all. The only real rule is that if you submit the first half of a level, you've committed to finishing the second half of someone else's... other than that, I thought I'd leave things relaxed and see what happens, since this whole idea is something of an experiment.
However, is it against the rules to tweak the first half that I'd be receiving, or offer tips to the person who'll be editing my first half?
Both of those sound OK to me. As the second participant is basically in charge of completing the level, they may want it to be in a cohesive, polished state that would benefit from tweaking existing content. Additionally, the ways in which a level can be half-finished might vary since they are open to some interpretation: a level could be playable from start to finish but lack any story content or graphical embellishments, in which case editing existing work would be unavoidable. You could reach out to the author of the first half and ask permission if you really wanted to.
Offering tips could be a good idea, especially if your work includes any complex scripting that could daunt the recipient. Direct contribution to your original level during the second week is discouraged, but after that what happens to the level is up to its authors.